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College textbooks may be in the top-ten of the worst things for sale, ever. It's not bad enough that the universe makes you feel worthless if you don't get a degree, and then laughs at you when you want a job. No, along the four- or five-year journey to your worthless diploma, they make you buy dozens of textbooks. The future has brought slight reprieve to the textbook problem - you can buy them online for cheap, get free shipping, and resell them for more than the snotty guy at the campus bookstore wants to give you when the class is over. But the fundamental issue remains that introductory calculus, or chemistry, or whatever, has not changed in at least twenty years. The only difference is the word problems have changed. Skrillex buys an ice cream cone whose height is h and radius r, topped with a sphere of ice cream with radius 1.1r. His friend Deadmau5 texts him on an iPhone 4, and while he texts back, the ice cream melts and runs into the cone. The cone has a leak which allows the melted ice cream to run out the bottom at rate 0.031r3 per minute (t). Express the surface area of the cone filled with melted ice cream as a function of time. Do not use rage faces in your solution.
In the context of very bright high school students with strong mathematics backgrounds, it is typical to teach discrete math to students without requiring calculus as a prerequisite. In particular, this is the norm both at the Ross program (where 2nd year students often had a combinatorics class) and at Mathcamp (where many discrete math classes are often taught without calculus as a prerequisite). Both summer programs avoid teaching calculus because it messes up highschool students who are going to be stuck taking calculus whether they already know it or not. In particular, it's quite possible to teach formal differentiation and integration of power series in order to do generating functions without discussing traditional differentation or limits. In fact, the Ross problem sets had a problem set developing the basics of calculus for polynomials (linearity, Leibniz rule, etc.) without ever discussing limits. I'd already learned calculus at that point, but not all the students had. And the students who didn't know calculus didn't have too much of a difficulty with that problem set. It's certainly easier than proving that the group of units modulo p is cyclic. So the reason for requiring such a prerequisite for a college course is not that it's actually a logical prerequisite, but instead for sociological reasons along the lines of Alex's answer.
@book {MATHEDUC.06068645, author = {Deiser, Oliver}, title = {First aid in analysis. Overview and basic knowledge with numerous illustrations and examples. (Erste Hilfe in Analysis. \"Uberblick und Grundwissen mit vielen Abbildungen und Beispielen.)}, year = {2012}, isbn = {978-3-8274-2994-0}, pages = {246~p.}, publisher = {Berlin: Springer Spektrum}, doi = {10.1007/978-3-8274-2995-7}, abstract = {This book takes up the analysis, as it is normally taught at the pre-university levels, which tends to be a descriptive way to introduce the concepts of analysis. In carefully chosen steps, the author translates these notions into a purely formalistic language, from the basics through to integration. Numerous examples and illustrations help the reader to understand this formal language. This then allows the author to present almost all exotic examples, which can hardly be explained in the descriptive language. The reader will note with comfort that the author also recommends not to forget the intuitive content of this formalism.}, reviewer = {Hansueli H\"osli (Ittigen)}, msc2010 = {I10xx (U20xx)}, identifier = {2013a.00673}, }
We are moving to our new website at nmss.edu.au soon, you can preview it now. The National Mathematics Summer School (NMSS) is a program for the discovery and development of mathematically gifted and talented students from all over Australia. It is a two-week residential summer course held each January at The Australian National University (ANU) in Canberra. Students participate in a series of lecture courses from mathematicians in a number of branches of mathematics at a relatively advanced level. They attend tutorials under the guidance of a range of staff — postgraduate students, mathematics teachers and academic mathematicians. The main activity of NMSS is an in-depth study of three or four different areas of mathematics. Each is very challenging and will extend every student. On the other hand, the program is non-competitive and very much hands-on. The emphasis is on doing mathematics, not just on listening to someone else talking about it. By the end of the two weeks, most students are amazed at how much they have accomplished and post-school surveys indicate that the NMSS has succeeded in raising their intellectual horizons. Almost everyone returns home with a considerably enhanced view of their own potential.
Materials and Resources Objectives Students will be able to solve linear systems and linear-quadratic systems algebraically. Given the steps of a systems problem students will be able to place them in the correct order. Duration One 45-minute class period Materials Solving Systems Algebraically SMART notebook lesson Solving Systems Algebraically Worksheet Extra Examples Worksheet SMART Board, Projector, computer Procedure Students will review methods for solving linear systems of equations and linear-quadratic systems of equations. Display "Solving Systems Algebraically SMART notebook file". Page One: Contains an example, with steps for the substitution method for solving linear systems. Each step is hidden under a white box that is labeled with a small number to indicate the order that the boxes should be moved. Students should copy this example into their notebooks. Page Two: One linear system example to be worked through with the students. Again, this example should go in their notebook. Page Three: Contains an example, with steps for the substitution method for solving linear-quadratic systems. Each step is hidden under a white box that is labeled with a small number to indicate the order that the boxes should be moved. Students should copy this example into their notebooks. Page Four: One linear-quadratic system example to be worked through with the students. Again, this example should go in their notebook. Page Five: Here is the first interactive example. This is to be completed by the teacher to demonstrate what students are expected to do in later slides. There are steps to the problem on the right side of the slide. Each step has been set to infinitely clone. The teacher should select the steps in the correct order, drag each and place it under the example. This is the first example on the worksheet "Solving Systems Algebraically". Students should complete the work there. Pages Six - Eleven: More interactive examples. Students should come to the board and follow the steps that the teacher presented in the first example. Students sitting in their seats should be completing the examples on the worksheet. The examples on the pages are in the same order as the examples on the worksheet. Page Twelve: Extra Examples. Students can work on these in class or it can be done for homework. These problems can also be found on the "Extra Examples Worksheet". Assessment Students understanding will be assessed upon completion of their SMART board problem. As well as, completion of each worksheet.
JWST Mathematics Core Content This page will connect you to the part of the course that deals with mathematics, its relation to the NASA missions and to problem sets to help you understand and expand your learning about these concepts.
MATHEMATICAL LITERACY (Bilingual) ISBN: 978-1-86863-056-1 Math Lit Grd. 11 & 12 This is a comprehensive study guide which will help Grd. 11 and 12 learners to understand and master all the basic concepts and procedures in the Mathematical Literacy Core Curriculum, an aspect often lacking in Mathematical Literacy textbooks. The study guide includes many exercises (with complete solutions) which allow learners to master these concepts and procedures in various contexts. The book follows a simple and structured approach. It deals with one Learning Outcome at a time, explaining and providing examples and exercises about fundamental topics such as:
Finite Mathematics and Its Applicationsused - like new 10
Courses - Numerical Analysis Lecturer: Sorin Pop The purpose of this course is to provide introductory material in numerical methods. The course is split into two parts, part A and part B. Part A is meant for trainees of all orientations, whereas part B is meant for trainees of the T-orientation. In part A basic numerical methods and algorithms are introduced and the main elementary techniques are discussed. Part B is dedicated to special techniques for solving ordinary and partial differential equations which apply to problems from mathematical physics, their modeling and numerical simulation. At the end of the course, the trainee should be able to find appropriate numerical techniques for certain problems, and be able to make a choice by analyzing the advantages and disadvantages of the available methods. Teaching is organized in weekly sessions of two hours. Students are expected to complete their knowledge by self-study. Assignments are given every week and have to be handed in the week after. Every two weeks the students are given a team project. Trainees entering the course are expected to have basic knowledge of analysis, linear algebra, ordinary and partial differential equations.
Mathematical Methods for Physicists This text is designed for an intermediate-level, two-semester undergraduate course in mathematical physics. It includes details of all important tools required in physics, and contains a large number of worked examples to illustrate the mathematical techniques developed and to show their relevance to physics. The advancement of observational techniques over the years has led to the discovery of a large number of stars exhibiting complex spectral structures, thus necessitating the search for new techniques ... Mathematical Tools for Physisists is a unique collection of 18 review articles, each one written by a renowned expert of its field. Their professional style will be beneficial for advanced students ...
Chapter 2: The Real Numbers Although you have doubtless worked quite a bit with the real numbers, this chapter will start at the beginning introducing you to them again from a perhaps higher viewpoint than that you have seen in the past. This will serve to put your knowledge in a proper mathematical framework so that you can better understand some of the topics of the last chapter. Here are some of the topics: We begin by defining a field; it is basically a set like the rational numbers or the real numbers where one can operate in the usual way with the four basic arithmetic operations. We then add in the idea of an order. This allows us to work with inequalities in the usual manner. The Real and rational numbers both have order relations. In order to distinguish between, say, the rationals and the reals, one needs one further property. This amounts to assuming that infinite decimals correspond to numbers. This will complete our definition of the real numbers. The set of integers is a particularly important set of numbers. In fact, we could spend the entire semester learning about them; for our purposes, we need to know that integers factor uniquely into products of primes. This result is called the Fundamental Theorem of Arithmetic and it is well worth understanding why it is true. The real numbers are not adequate to guarantee that every polynomial has a root. To get this property, one needs the field of complex numbers. The basics of complex numbers will be introduced. A set is simply a collection of objects. We could define axioms for set theory, but, instead, we choose to depend on your intuitive understanding of sets. What can you do with a set? Basically, you can check to see if an object is in the set (we then say that the object is an element of the set). You can also define subsets of the set; a subset is simply a collection of objects, every one of which is in the original set. Every set contains the empty subset (the collection with no elements in it), and the original set itself (the collection of all elements of the original set). Furthermore, we assume you are familiar with basic operations like the union and intersection of subsets of a set. Given two sets S and T we can define the cartesian product of S and T as the set whose elements are all the ordered pairs (s, t) where s is in S and t is in T. For example, in the last chapter, we defined took the cartesian product of the set of real numbers with itself as the set of points in the plane. By a relation between the sets S and T, we mean simply any subset of the cartesian product . This may seem like a strange definition. But it is really what we mean by a relation. For example, the property of two real numbers a and b that a be less than b is a relation. It is a relation between the set of real numbers and itself, and it is completely determined by specifying the set of all pairs of numbers (a, b) such that . As another example, we have the relation of b being the mother of b. Let S be the set of all people and T be the set of all women. Then this relation is simply the set of all pairs (s, t) in where t is the mother of s. A function from the set S to the set T is simply a relation with the property that for every s in S there is exactly one element in R with first coordinate s. The set S is called the domain of the function and set T is called the range space of the function. We often indicate that R is a function with domain S and range space T by using the notation: . The set of all elements t in T for which there is at least one pair in R with second coordinate t is called the range of the function. We write R(a) = b if (a, b) is in R. Example 1: Let N be the set of positive integers. The successor function is the function with s(k) = k + 1. The function is the set of all pairs (k, k + 1) where k is a positive integer. The domain of s is N, its range space is N, but it range is the set of integers greater than or equal to 2. Let R be the set of real numbers. The set of all pairs where x is in R is a function. We normally write it as . Let R be the set of real numbers. The set of all pairs where x is in R is not a function from R to R. This is because there is no pair with first coordinate a negative number. You could consider it to be a function of from the set of non-negative real numbers to R. Let S be the set of non-negative real numbers. The set of all pairs (a, b) where is not a function. For example, (4, 2) and (4, -2) are two pairs in the set with the same first coordinate. Warning: Many textbooks define a function f from S to T as a rule which assigns to each element s in S exactly one element t in T. They then say that the set of all points (s, f(s)) is called the graph of the function. Although this approach corresponds more closely to one's intuitive notion of a function, we have avoided this approach because it is not clear what one means by a rule. In the case where the domain and range space are both the set R of real numbers, we see that the function is the set of points in what we would have called the graph of the function. The condition that there be exactly one point in the graph with an given first coordinate amounts to saying that every vertical line intersects the graph in exactly one point. For this reason, the condition is often referred to as the vertical line test. A binary operator on a set S is a function from to S. A unary operator on a set S is a function from S to S. Example 2: Addition and Multiplication on the real numbers are both binary operators. Negation is a typical unary operator. Binary operators are usually written in infix notation like 5 + 7 rather than in functional notation like +(5,7). Similarly, unary operators are usually written in prefix notation like -3 rather than functional notation like -(3). The collection of elements out of which we will be making algebraic expressions will be referred to as a field. More precisely, a field is set endowed with two binary operators which satisfy some simple algebraic properties: Definition 1: A field F is a consisting of a set S and two binary operators called addition and multiplication which satisfy the following the properties: (Commutativity) For every a and b in S, one has a + b = b + a and ab = ba. (Associativity) For every a, b, and c in S, one has (a + b) + c = a + (b + c) and (ab)c = a(bc). (Distributive Law) For every a, b, and c in S, one has a(b + c) = ab + ac. (Identities) There is an element 0 in S such that for all a in S, one has a + 0 = a. There is an element 1 different from 0 in S such that for all a in S, one has acdot 1 = a. (Inverses) For every a in S, there is an element denoted -a in S such that a + (-a) = 0. For every a in S other than 0, there is an element denoted such that . We usually will denote the set S with the same symbol F as is used for the field. Example 3: The set of rational numbers with the usual addition and multiplication operators is a field. The same is true of the set of real numbers with the usual addition and multiplication operators. The set of complex numbers (yet to be defined) will also be a field. Example 4: Consider the set of expressions of the form p(x)/q(x) where p and q are polynomials with real coefficients and q is non-zero. We say that if (where the multiplication is the usual multiplication of polynomials). Addition and multiplication of these expressions are defined in the usual manner. One can show that these expressions form a field called the field of rational functions. Warning: We are treating rational functions here as simply expressions, not as functions. In particular, the definition of equality does not correspond to equality of functions. Example 5: There is basically only one way to make a field with only two elements 0 and 1. See if you can make up the appropriate addition and multiplication table and verify the field properties. The definition of a field includes only the most basic algebraic properties of addition and multiplication. We will see, however, that all the usual rules for manipulating algebraic expressions are consequences of these basic properties. First, let use begin by noting that the definition of a field only assumes the existence of identities and inverses. In fact, it follows that they are in fact unique: Proposition 1: If F is a field, then the identity elements 0 and 1 as well as the additive and multiplicative inverses are unique. Proof: Suppose that 0 and 0' are additive identities. Then, since 0 is an identity, we have 0' + 0 = 0'. Similarly, since 0' is an identity, we have 0 + 0' = 0. Since addition is commutative, we conclude that 0 = 0'. Let a be in the field. Suppose that both b and c are additive inverses of a. The a + b = 0 and a + c = 0. We can now calculate b = b + 0 = b + (a + c) = (b + a) + c = (a + b) + c = 0 + c = c + 0 = c. (Please be sure that you understand why each of the steps of this calculation are true.) To complete the proof, you should make similar arguments for multiplicative identities and inverses. Proposition 2: Let F be a field. If a is in F, then . If a and b are in F satisfy , then either a or b are zero. Proof: Let a be in F. Then . Let b be the additive inverse of . Then applying it to the last equation, we get Suppose ab = 0. If a is zero, there is nothing more to prove. On the other hand, if , then a has a multiplicative inverse c and so . Corollary 1: If a is an element of a field F, then -a = (-1)a. Proof: . We can define the binary subtraction operator: a - b = a + (-b) and, for , the binary division operator . The division operator will also be expressed as . Proposition 3: Let F be a field containing a, b, c, and d where b and d are non-zero. Then If c is also non-zero, then Proof: (i) Do this as an exercise - it is a matter of simplifying using the commutative and associativity properties of multiplication to see that the product is equal to 1. (ii) Using associativity and commutativity, one can show that . By assertion (i), this is the same as . (iii) (iv) One has where some of the steps have been combined. (v) Using property ii, it is easy to see that . But then assertion v follows from assertion ii. This completes the proof of the proposition. The remaining rules in section 1.1.1 on simplifying expressions are now easy to verify, i.e. they are properties of any field. The material in section 1.1.2 and 1.1.3 on solving linear equations or systems of linear equations are also properties of fields. On the other hand, the material on solving quadratics does not hold for arbitrary fields, both because it uses the order relation of real numbers as well as the existence of square roots. Definition 2: An ordered field F is a field (i.e. a set with addition and multiplication satisfying the conditions of Definition 1) with a binary relation < which satisfies: (Trichotomy) For every pair of elements a and b in F, exactly one of the following is true: a < b, a = b, and b < a. (Transitivity) Let a, b, c be arbitrary elements of F. If a < b and b < c, then a < c. If a, b, and c in F satisfy a < b, then a + c < b + c. If a, b, and c in F satisfy a < b and 0 < c, then ac < bc. Fact: If F is an ordered field, then 0 < 1. Proof: By Definition 1, and so by trichotomy, if the the fact were wrong, then we would have a field F with 1 < 0. By property iii, we would have 1 + (-1) < 0 + (-1) and so 0 < -1. But then using property iv, we would have . By Proposition 2, the left side is 0 and so . This contradicts trichotomy and so the assertion must be true. If F is an ordered field, an element a in F is called positive if 0 < a. Proposition 4: The set P of positive elements in an ordered field F satisfy: (Trichotomy) For every a in F, exactly one of the following conditions holds: a is in P, a = 0, and -a is in P. (Closure) If a and b are in P, then so are a + b and ab. Proof: (i) By property i of the Definition 2, exactly one of a < 0, a = 0, and 0 < a must be true. If a < 0, then by property iii of Definition 2, we have a + (-a) < 0 + (-a) and so 0 < -a. Conversely, if 0 < -a, adding a to both sides gives a < 0. So the three conditions are the same as -a is in P, a = 0, and a is in P. Suppose a and b are in P. Then 0 < a and by property iii of Definition 2, we have 0 + b < a + b and . Since 0 < b and b < a + b, transitivity implies that 0 < a + b. Since by Proposition 2, we have 0 < ab. Remarks: i. In one of the exercises, you will show that, if a field has a set P of elements which satisfy the conditions of Proposition 4, then the field is an ordered field assuming that one defines a < b if and only if b - a is in P. ii. An element a of an ordered field F is said to be negative if and only if a < 0. iii. It is convenient to use the other standard order relations. They can all be defined in terms of <. For example, we define a > b to mean b < a. Also, we define to mean either a < b or a = b and similarly for . iv. The absolute value function is defined in the usual way: v. One now has everything you need to deal with inequalities as we did back in section 1.1.4. Proposition 5: Let a and b be elements of an ordered field F. \|-a\| = \|a\| (i.e. and ) (Triangle Inequality) Proof: i. By Trichotomy, we can treat three cases: a > 0, a = 0, and a < 0. If a > 0, then -a < 0 and so \|a\| = a and so \|-a\| = -(-a) = a. If a = 0, then -a = 0 and so \|a\| = 0 = \|-a\|. If a < 0, then -a > 0 and so \|a\| = -a and \|-a\| = -a. In all three cases, we have \|a\| = \|-a\|. ii. Again, we can treat three cases: If a > 0 or a = 0, then \|a\| = a and so . If a < 0, then adding -a to both sides gives 0 < -a and so a < -a by transitivity. In this case we have \|a\| = -a and so a < \|a\|. We could argue the other inequality the same way, but notice that we could also use our result replacing a with -a. (Since it holds for all a in F, it holds for -a.) The result says , where we have used assertion i. Adding a - \|a\| to both sides of the inequality gives the desired inequality. iii. Once again, do this by considering cases: If , then \|a + b\| = a + b. Since and , we can add b to both sides of the first inequality and \|a\| to both sides of the second one to get and . Using transitivity, we get as desired. Now suppose that . Then adding -a - b to both sides of the inequality gives -a + (-b) < 0. Applying the result of the last paragraph, we get . But a + b < 0 means that \|a + b\| = -(a + b) and so where we have used assertion i for the last step. This completes the proof. Example 6: Let a, b, c, and d be elements of an ordered field F. Then ( Cauchy -Schwarz Inequality). This is an example of how to deal with a more complicated inequality. It is not clear how to begin. In such a case, it is often useful to simply work with the result trying to transform it into something easier to prove. So, suppose the result were true. We could then expand out the expression using the distributive law to get: which simplifies to . This still looks complicated until one thinks of grouping the factors differently to get . Subtracting the left side from both sides of the inequality gives . Now, you may recognize the right side as being a perfect square: factoring we get . It is still complicated, but do you expect that it is true? The right hand side is the square of a complicated expression -- and the assertion is that it is non-negative. We have not yet proved this, but it is a common property of real numbers, so you might make the Conjecture: If a is any element of an ordered field, then . The conjecture looks like it should be simple enough to prove. In fact, you should go ahead and try an prove by considering cases as we have done before. Once it is proved, is the Cauchy-Schwartz Inequality proved? No. We had assumed that it was true and shown that it a result which would be true provided the conjecture were true. That is no proof of the inequality. But, all is not lost, the idea is that we might be able to trace back through our steps in reverse order and reach the desired inequality. Assuming that the conjecture is true, let's see that the Cauchy-Schwartz inequality must also be true. We know that the square of any field element is non-negative. So, applying this to the element , we get . Using the distributive law to expand this gives . Adding 2(ad)(bc) to both sides of the inequality yields . Using the commmutative and associative law several times allows us to re-arrange this into . Adding appropriate terms to both sides and again using associativity and commutativity takes us back to the step: Finally, we can factor the sides to get the Cauchy-Schwartz Inequality. Only later, will you see that the Cauchy-Schwartz inequality is useful. But already there is a lesson here. If we had just presented the last paragraph as a proof, you would have no idea of why one was doing each of the steps -- you would know that the inequality was true, but have no intuitive grasp of why or how one ever came up with the proof. In this case, the idea is that the proof is obtained by non-deductive means -- we simply worked backward from the result we wanted to prove until we got to an assertion we could prove. We proved the assertion and then used it to work forward to the desired result. When reading any proof you should always be asking yourself whether or not the proof is something you could have come up with yourself. If not, then you need to work more with the material until you hopefully will understand enough to be able to do it yourself. The so-called natural numbers (Are the others un-natural?) are the numbers 1, 2, 3, etc. But expressing this is a bit complicated. Assume for the whole section that we are operating within a particular ordered field F. The set S we want to describe satisfies the conditions: , 1 is in S, and If a is in S, then so a + 1. Any such set S will be called inductive. Now consider the set T of all inductive sets S. For example, the set F is in T as well as the set of all positive elements of F. The set of natural numbers would appear to be contained in any of the sets in T. So, one way to define the set we want is Definition 3: The set of natural numbers is the intersection of all inductive sets, i.e. a is a natural number provided that it is an element of every set S in T. Proposition 6: (Induction) The set of natural numbers satisfy the conditions , 1 is in , and If a is in , then so a + 1. The only inductive subset of is . Proof: i. Let a be in . Then a is an element of every set S in T. If S is in T, then an so a is an element of F. ii. Since 1 is in every set S which is in T, 1 is in their intersection which is, by definition, . iii. Let a be in . If S is in T, then a must also be in S. But then a + 1 is also in S. Since this is true for every S in T, a + 1 is in the intersection of all the S in T, i.e. a + 1 is in . iv. Suppose S is an inductive subset of . Then S is in T. Since is the intersection of all the S in T, it follows that is contained in S. But then we have and , which means that the two sets must be equal. This completes the proof. The importance of Proposition 6 is that it is the basis of a method of definition and of proof called mathematical induction which we will normally refer to simply as induction. First, let's see how it works for definitions. Let's do an inductive definition of powers. Let a be in F. We define to be a. Now, suppose we have already defined for some natural number k, then define to be the product . Consider the set S of all natural numbers k for which we have defined the . It contains 1 and if k is in S, so is k + 1. By Proposition 6, it follows that is defined for all natural numbers k. We can also use induction in proofs. Here is the general scheme: Suppose that for every natural number k, we have a property P(k). Assume furthermore that: P(1) is true. For every natural number k, if P(k) is true, then so is P(k+1). Then the set of natural numbers for which P(k) is true must be the set of all natural numbers since it is inductive. So, P(k) is true for every natural number k. Example 7: Let's prove that whenever m and n are natural numbers. We use induction with the property P(k) being the condition on k that for all natural numbers m, we have . For k = 1, we have by the inductive definition of the powers of a. So P(1) is true. Suppose that we know that P(k) is true for some particular k. Let m be a natural number. We have For the record, the justification for each of the steps in the above series of equalities is: inductive definition of powers associativity of multiplication P(k) inductive definition of powers associativity of multiplication So, P(k + 1) is true. By Proposition 6, it follows that the P(k) is true for all natural numbers k. Suppose we wanted to show the same result for all integers n which greater or equal to zero. There are two possibilities: either show the special case of n = 0 separately. Or, you could define P(k) to be the property which we were calling P(k - 1); either approach is equally valid. The set of integers is defined to be the set of all elements in F which are either natural numbers, 0, or whose negative is a natural number. If P(k) is defined for all integers k, then you can sometimes prove that P(k) is true for all integers k by using two induction proofs, the first showing that it is true for all non-negative integers and the second showing that it is true for all negative integers. Warning: From the exercises, you will see that proof by induction is an extremely powerful tool. If one is dealing with inductively defined quantities like positive integer powers of a number, then induction is both natural and leads to a good understanding. On the other hand, it is often the case that even though you can prove things by induction, you are left with the feeling that you still do not have any intuitive understanding of why the result should be true. So, whereas induction may lead to a quick and easy proof, the result can be less than fully satisfying. Lemma 1: for every natural number b. Proof: Let P(k) be the property that . Clearly P(1) is true. If for some natural number k, we have P(k) true, then . Since we have already seen that 1 is positive, we have 0 < 1. Adding k to each side, we get k = k + 0 < k + 1. By transitivity, it follows that , and so P(k+1) is true. Therefore, P(k) is true for all natural numbers k. Lemma 2: If k is a natural number there is no natural number m with k < m < k + 1. Proof: Suppose that there is a natural numbers n and m with n < m < n + 1. Let S be the set of all natural numbers except m. Then S is a subset of the set of natural numbers and 1 is in S. (If 1 were not in S, then we would have to have 1 = m because m is the only natural number not in S; but then n < m = 1 contrary to Lemma 1.) Further, if k is any natural number in S such that k + 1 is not in S, then k + 1 = m since m is the only natural number not in S. But then n < m = k + 1 < n + 1 implies n - 1 < k < n. So there is a natural number lying strictly between n - 1 and n. Now let P(k) be the property that there be no natural number m lying strictly between k and k + 1. Proceeding by induction, let us note that P(1) is true. If not, then letting n = 1 in the last paragraph, we see that 0 = n - 1 < m < n = 1. This contradicts Lemma 1 since m is a natural number. Now suppose that P(k) is true but P(k + 1) is false. So there is a natural number m with k + 1 < m < k + 2. But the result of the first paragraph of the proof with n = k + 1 then shows that there is a natural number lying strictly between k and k + 1. But this contradicts the assumption that P(k) is true. We conclude therefore that if P(k) is true, then so is P(k + 1). By induction, it follows that P(k) is true for all natural numbers k, and so Lemma 2 is proved. Proposition 7: (Descent) Every non-empty set S of natural numbers contains a smallest element, i.e. there is an a in S such that for all b in S. Proof: Suppose S is a non-empty set of natural numbers that does not have a smallest element. Let S' be the set of all natural numbers smaller than all elements of S. The element 1 must be in S' because otherwise 1 would be the smallest element of S by Lemma 1. Let k be any natural number in S' such that k + 1 is not in S'. Since k + 1 is not in S', there must be a natural number n in S for which . Now n must be greater than k since k is in S'. We cannot have k < n < k + 1 by Lemma 2 and so , which means that n = k + 1. We will show that n is the smallest element in S. For suppose m is any element of S. We have k < m because k is in S'. We cannot have k < m < k + 1 = n by Lemma 2. So we have . So n is the smallest element of S. Since we assumed that S had no smallest element, we have a contradiction. This proves Proposition 7. The value of Proposition 7 is that it is the basis for another proof technique called infinite descent which is another variant on induction. Here is an example. Example 8: Prove that for every a in F and for every natural number n and m, we have: . Let P(k) be the property that for all a and m. If P(k) were not true for all natural numbers k, then the set S of natural numbers for which it were false would be non-empty. By Proposition 7, there is a smallest natural number k for which P(k) is false. Now, k cannot be 1, since . So k - 1 must also be a natural number and P(k) must be true (otherwise k would not be the smallest element of S). So . But then contrary to assumption. Definition 4: A natural number d is a divisor of the natural n if and only if there is a natural number m such that n = dm. A divisor d of n is said to be proper if it is other than 1 and n itself. A natural number p is said to be prime if it is not equal to 1 and it has no proper divisors. For example, 4 is a divisor of 20 because . The prime numbers are 2, 3, 5, 7, 11, etc. Let us define by induction the product of n numbers. A product of 1 number is defined to be itself. Assuming that we have defined the product of k numbers. A product of k + 1 numbers is the product of the product of the first k of the numbers and the last number. For convenience, we also stipulate that the product of zero numbers is 1. If for are n numbers, then the product of these n numbers is denoted . Of course, if all the numbers are equal to a, we can still denote the product of a with itself n times as . (One defines the sum of n numbers in an analogous manner; the notation for the sum of the n numbers for is .) ii. Every natural number can be written as a product of finitely many primes. Proof: i. Suppose dm = n. Then and so . We also have 1 leq d since d is a natural number. So, . In particular, if d is a proper divisor of n, then 1 < d < n. ii. We will prove the second assertion by infinite descent. If the assertion is false, then there is a smallest natural number n which is not expressible a product of primes. Then n is not a prime or else it would be the product of a single prime. Since n is not a prime, we can write n = dm where d is a proper factor of n. But then m is also a proper factor of n. (Why?) In particular, both d and m are natural numbers smaller than n. Since n was the smallest number not expressible as a product of primes, both d and m can be expressed as a product of primes. But then by taking the product of all the factors in the expressions of both d and m, we see that n is a product of primes also. This contradiction proves the result. Proposition 9: (Division Theorem) If n and d are natural numbers, there there are unique non-negative integers m and r such that n = md + r and . Proof: Let us prove the result by induction on n. If n is 1 and d is also 1, then m = 1 and r = 0 is the unique solution. On the other hand, if n = 1 and d > 1, then m = 0 and r = 1 is the unique solution. Now, suppose that the result is true for some natural number n. Then n = md + r with . If r = d - 1, then n + 1 = (m+1)d + 0. Otherwise, n + 1 = md + (r + 1). To show uniqueness, suppose that with for i = 1 and 2. Taking differences, we see that where we have ordered the terms so that the right hand side is non-negative (If it were not, just swap the subscripts). But then the left side is also non-negative. So, we must have . Since the left side is a multiple of d, Proposition 8 implies that it must be zero. So . But then, the right side is also zero and so too. If m and n are natural numbers, then a natural number d is called a common divisor of m and n if it is both a divisor of m and also a divisor of n. The largest common divisor of m and n is called the greatest common divisor of m and n and is denoted gcd(m, n). Proposition 9 will allow us to develop the Euclidean Algorithm for calculating gcd(m, n). Repeatedly apply Proposition 9 to obtain where one stops with as soon as one obtains the first zero remainder. If d is a common divisor of m and n, then d also divides by the first equation. But then, the second equation shows that d divides r_1, and so on. We finally determine that d divides all the remainders. In particular, it divides the last remainder . On the other hand, starting from the last equation, we see that divides . The second from the last equation then says that it also divides , and so on. We conclude that divides both n and m. We conclude therefore that . Again starting from the second from the last equation, we can solve to get . Using the previous equation we can solve for and substitute the expression into the right hand side of this last equation to express as a linear combination of and . Repeating the process, we eventually get the greatest common denominator written as a linear combination of m and n. Summarizing the results: Proposition 10: (Euclidean Algorithm) If m and n are natural numbers, the Euclidean algorithm described above calculates the greatest common divisor of m and n. Furthermore, it allows one to find integers a and b such that gcd(m, n) = am + bn. We say that two natural numbers m and n are relatively prime if gcd(m,n) = 1. Corollary 2: If two natural numbers m and n are relatively prime, then there are integers a and b with 1 = am + bn. In particular, if p is a prime and n is not a multiple of p, then p and n are relatively prime and so there are integers a and b with 1 = ap + bn. Corollary 3: If a prime p divides a product mn of natural numbers, then either p divides m or p divides n. Proof: If it divides neither, then there are integers a, b, c, and d such that 1 = ap + bm and 1 = cp + dn. Taking products, we get 1 = 1cdot 1 = (ap + bm)(cp + dn) = (acp + adn +bmc)p + bd(mn). Since p divides mn, we have mn = ep for some e. Substituting, we see that p divides the right side and so p must divide the left side. But the left side is 1, which is a contradiction. So, it must be that p divides m or p divides n. Corollary 4: If a prime p divides a product of any finite number of natural numbers, then p divides at least one of the numbers. Proof: This follows by induction from Corollary 3. Corollary 5: (Linear Diophantine Equations) The equation ax + by = c where a, b, and c are constants has integer solutions (x, y) if and only if gcd(a,b) divides c. Proof: If a or b is zero, then the result is obvious. If not, then we can assume that a and b are natural numbers (by replacing one or both of x and y with their negatives). If gcd(a,b) divides c, then the Euclidean Algorithm gives integers u and v with au + bv = gcd(a,b). Multiplying by d = c/gcd(a,b) shows that x = ud and y = vd are solutions of the equation. On the other hand, if there is a solution, then clearly gcd(a,b) divides ax + by = c. Example 9: Let's calculate the gcd(310, 464). One has , , and . We conclude that gcd(310, 464) = 2. Furthermore, we have and . Substituting gives . Theorem 1:(Fundamental Theorem of Arithmetic) Every natural number can be represented as a product of zero or more primes. Furthermore, these primes are uniquely determined (including the number of times each prime is repeated) up to order of the factors. Proof: We already know that at least one representation exists. If the result is false, then let n be the smallest natural number for which uniqueness is false. Suppose one can write two distinct representations where the factors are all prime. There cannot be a prime p which appears in both factorizations; if there were, then n/p would be a smaller natural number with two distinct representations as products of primes. But, clearly is a factor of n. Corollary 4 implies that must divide some . Since is a prime, it follows that as primes have no proper divisors. This contradiction shows that there is no natural number n for which the factorization is non-unique. This completes the proof. Proof: Suppose that the only prime numbers were . Let n = . Clearly n is a natural number not divisible by contrary to the Fundamental Theorem. Corollary 7: is irrational. Proof: Suppose that where m and n are positive integers. By dividing m and n by their greatest common divisor, we may assume that they are relatively prime. Squaring both sides of the equation and multiplying through by the denominator, we get . Since 2 divides the left side, 2 divides and, since 2 is a prime, we have 2 divides m. Dividing our equation by 2, we get . So 2 divides the right side and is divisible by 2. Again, this means that 2 is a divisor of n. But this contradicts the assumption that m and n are relatively prime. So, our original assumption that is rational must have been false. This completes the proof. All of the material in the previous sections of this chapter applies to any ordered field. In particular, it applies to both the fields of rational numbers as well to the fields of real numbers. The problem with working in the field of rational numbers is that it is relatively sparse; so, when you go to solve equations of degree greater than one, we often find that what would have been a solution is not rational. We have already seen that is not a rational number. It will be convenient to have an algebraic domain in which every polynomial equation has a solution. We will find that the complex numbers fill this role; the real number field will be both useful to construct the complex numbers as well as being important in and of itself. Exactly what makes the real numbers special is a rather subtle matter. This section will start that explanation, and the version given here will suffice until we can revisit the question in a later chapter. In section 1.2.2, we said that real numbers could be represented as infinite decimals. This is the aspect of real numbers that we will discuss in this First let us start with a a finite decimal. This is a numeral of the form 3.14159. The form of a finite decimal is an optional sign (either plus or minus) followed by a string of decimal digits, a period, and another string of decimal digits. Each of these represents a rational number: If there are n digits to the right of the period (called the decimal point), then the rational number is the quotient of the decimal (with the period removed) divided by . For example, we have . Because the denominator is always a power of 10, many rational numbers cannot be represented as a finite decimal. For example, 1/3 cannot be written as a finite decimal. On the other hand, we can write arbitrarily good approximations: 0.3, 0.33, 0.333, 0.3333, etc. of 1/3. So, it is reasonable to say that if we just allowed ourselves to keep writing digits, we would get 1/3. You might write this as 0.333333.... where the ellipsis means to keep repeating the pattern. Another example would be 1/7 = 0.142857142857142857.... These are examples of infinite decimals. But in what sense do these represent the rational number? To get a better idea, let's look again at how we convert a finite decimal to a fraction. If the decimal is , then the part is an integer, the digit means , the digit is in the next place and it represents , and so on. So our whole decimal becomes Let's go back to our specific examples, we have a succession of improving estimates: where there are n digits 3 in the last approximation. Our last one is then It is still hard to tell what value we are approaching as n gets larger and larger. Here is the secret to calculating the sum: Lemma 3: (Geometric Series) If , then Proof: This is actually a familiar factorization of which the first few cases are: and . To understand how this works, just multiply the left side of our general expression by (1 - a). By the distributive law, this is the same as the original expression less the product of a and the original expression, i.e. . Notice that almost all the terms are repeated in the second expression. The one left out is 1 and we have one additional one . So the difference is just , which shows the result. Applying Lemma 3 with a = 1/10 gives and the right side simplifies to . This is the exact value when there are precisely n digits to the right of the decimal point. Now, what happens when we take more and more digits? The result is always a little less than 1/3, but the error shrinks to zero as n gets arbitrarily large. This is the sense in which we can say that the infinite decimal represents 1/3. Let's repeat the same computation with our second example. In this case, it is inconvenient to use powers of 10 because the pattern repeats itself every 6 digits. But things are easy if we simply use powers of . where again we have repeated the pattern exactly n times. Lemma 3 says that this is equal to Clearly, as n gets arbitrarily large the right factor approaches 1. Furthermore, if you reduce the fraction, you will see that 142857/999999 = 1/7. Again, we see that the infinite decimal represents 1/7 in the sense that if we take the sequence of numbers we get by taking more and more digits, the limiting value of the elements of the sequence is 1/7. Now, let's formalize our discussion. Definition 5: i. An infinite decimal is an expression of the type , where is an integer, and is an infinite sequence of decimal digits (i.e. integers between 0 and 9). ii. Every such infinite decimal defines a second sequence of finite decimals where . iii. One says that the infinite decimal represents the number r (or has limit r) if can be made arbitrarily close to zero simply by taking k sufficiently large. Definition 6: i. An ordered field F is said to be Archimedean if, for every positive a in F, there is a natural number N with a < N. ii. An Archimedean ordered field F is called the field of real numbers if every infinite decimal has a limit in F. Given any element a in F, we can form an infinite decimal for a. First, we can assume that a is positive, since the case where a = 0 is trivial, and if a < 0, then we can replace a with -a. Next, we see why we needed to add the Archimedean property to the above definition. Without it, we would not know how to get the integer part of a: Since F is Archimedean, the set of natural numbers N with a < N is non-empty and so there it has a smallest element b. Let . Then if . Choose to be the decimal digit such that and let , so that again . Assuming that we have already defined for some natural number k, the quantities and with , define by induction the digit so that and let , so that . The infinite decimal was defined so that with . So this infinite decimal has limit a. We say that this is the infinite decimal expansion of the element a in F. Proposition 11:i. Every element a in F is the limit of the infinite decimal expansion of a. ii. The decimal expansion of every rational number is a repeating decimal, i.e. except for an initial segment of the decimal, the decimal consists of repetitions of a single string of digits. iii. Every repeating decimal has limit a rational number. Proof: The first assertion has already been proved. For the second assertion, note that the definition of the sequence of digits is completely determined by the value of . If a = r/s is rational with r and s integers, then is a rational number with denominator (a factor of ) s. Furthermore, since , if is rational with denominator s, then so is . By induction, it follows is rational with denominator s for every k. Since lies between 0 and 1 and is rational with denominator s, it follows that there are at most s possible values for . The following principle is called the pigeonhole principle: If s + 1 objects are assigned values from a set of at most s possible values, then at least two of the objects must be assigned the same value. By the pigeonhole principle, there are subscripts i and j with such that . As indicated at the beginning of the proof, it follows that the sequence of digits starting from must be the same as the sequence of digits starting from and so the decimal repeats over and over again the cycle of values . The third assertion is easy to prove -- it is essentially the same as our calculation of the limit of the infinite decimal expansions of 1/3 and 1/7. The formalities are left as an exercise. Example 10: The field of real numbers contains many numbers which are not rational. All we need to do is choose a non-repeating decimal and it will have as its limit an irrational number. For example, you might take where at each step one adds another zero. Proposition 12: Every a > 0 in the field of real numbers has a positive -root for every natural number n, i.e. there is a real number b with . Proof:It is easy to show by induction that, if , then for every natural number n. So the function is an increasing function. By the Archimedean property, we know that there is a natural number M > a. Again by induction, it is easy to see that . By descent, it follows that there is a smallest natural number m such that . Let so that the -root of a must lie between and . Next evaluate for integers j from 0 to 10. The values start from a number no smaller than a and increase to a number larger than a. Let be the largest value of j for which the quantity is at most a. Repeating the process, one can define by induction an infinite decimal such that the -power of the finite decimal differs from a by no more than . Let b be the limit of the infinite decimal, and be the values of the corresponding finite decimals. Then we have and and so it is reasonable to expect that . This is in fact true. Using the identity for geometric series, we see that: . But then the triangle inequality gives where C is a positive constant which does not depend on k. Since this holds for all positive integers k, it follows that . A sequence of real numbers is said to converge to a real number b (or to have limit b) if all the are as close to b as desired as long as k is sufficiently large. More formally, this means that for every (regardless of how small), there is a (possibly quite large) N > 0 such that for all k larger than N. The sequence is said to be bounded above by a real number B if for all . For example, if , is an infinite decimal, and for , then converges to . Proposition 13: Let be a sequence of real numbers bounded above by a real number B. If the sequence is increasing, i.e. , then the sequence converges to some real number b. Proof: Since each is a real number, it has an decimal expansion where is a sign, either plus or minus. Because the sequence is increasing, one has: Except for a finite number of terms, the signs must all be identical. (Either they are all +, all -, or change once from - to plus.) Assume that the signs are all identical, say with value s, for Because there are only finitely many integer values between and B, the values of must all be identical for all for some which we can assume is no smaller than . Let be this common value. Assume by induction that we have defined and that for all for some no smaller than . Since the sequence is increasing, the for must be increasing (if s is +) or decreasing (if s is -). So, except for a finite number of terms, the numbers must be constant, say equal to for for some no smaller than . Let Then for all because the decimal expansion of all such b_i agrees with that of b up to the decimal digit. So b is the limit of the . This completes the proof. Remark: A sequence is said to be bounded below by a real number B if all the terms of the sequence are no smaller than B. A decreasing sequeence of real numbers which is bounded below converges to a real number. (To see this, simply apply the Proposition to the sequence .) Corollary 8: i. Let for be closed intervals with . If the lengths converge to 0, then there is precisely one real number c contained in all the intervals such that the sequence of the as well as the sequence of the converge to c. ii. Let the form a decreasing sequence of positive real numbers which converge to zero. Define for . Then the sequence converges to a real number b. (The value b is said to be the limit of the infinite sum . Proof: i. The sequence of the is increasing and bounded above by every . So, the sequence converges to a real number a which is no larger than any of the . Clearly, for all . Similarly, the sequence of the is decreasing and bounded below by every . So, the sequence converges to a real number b no smaller than any of the , and for all . We cannot have or else the distance between them would be positive; but this cannot be true because both lie in for all j and the sequence of the converges to zero. If a is a positive element of any ordered field, we know that because the set of positive numbers is closed under multiplication. Since we also have , it follows by trichotomy that the square of any element in an ordered field is always non-negative. In particular, such a field cannot contain a solution of . We would like to have a field where all polynomial equations have a root. We will define a field called the field of complex numbers which contains the field of rational numbers and which also has a root, denoted i, of the equation . In a later chapter, it will be shown that, in fact, contains a root of any polynomial with coefficients in . This result is called the Fundamental Theorem of Algebra. Let us first define the field of complex numbers. Since it is a field which contains both the field of real numbers and the element i, it must also contain expressions of the form z = a + bi where a and b are real numbers. Furthermore, there is no choice about how we would add and multiply such quantities if we wanted the field axioms to be satisfied. The operations can only be: and where we have used the assumption that . It is straightforward, but a bit tedious to show that these operations satisfy all the field axioms. Most of the verification is left to the exercises. But let us at least indicate how we would show that there are multiplicative inverses. Let us proceed heuristically -- we would expect the inverse of a + bi to be expressed as but this does not appear to be of the desired form because there is an i in the denominator. But our formula from geometric series shows how to rewrite it: We have . This is just what we need: Of course, we have proven nothing. But we now have a good guess that the multiplicative inverse might be . It is now an easy matter to check that this does indeed work as a multiplicative inverse. Proposition 14: The set of all expressions a + bi, where a and b are real and i behaves like , is a field if we define operations as shown above. We have already seen that the field cannot be ordered. Nevertheless, we can define an absolute value function by . Proposition 15: Let w and z be complex numbers. Then \|w\| = \|-w\| \|wz\| = \|w\|\|z\| \|z\| = 0 if and only if z = 0. . If r is a real number, its absolute value is the same as a real number as it is if it is considered to be the complex number . We have defined a 1-1 correspondence between the set of complex numbers and the Euclidean plane of pairs of real numbers. The complex numbers are not just a set; they also have addition and multiplication operators. Our next job is to see how these arithmetic operators correspond to geometric operations in the plane. The Parallelogram Law Let's start with addition. If for j = 1, 2, then . In our 1-1 correspondence we have the four numbers corresponding to the four points , , , . Here is a picture of the situation. From the picture, it certainly looks like the four points are vertices of a parallelogram. Showing that it is true is a simple matter of calculating the slopes of the four sides. For example, the slope of the line containing z_1 and z_1 + z_2 is which is the slope of the containing O and z_2. (You need to treat the case where separately: in this case, the two sides are coincident.) This result is the so-called parallelogram rule, which may be familiar to you as addition of forces in physics. Part of multiplication is easy: In Proposition 15, we have already seen the property where the absolute value is clearly just the length of the line segment from O to z (by the distance formula, a.k.a. Pythagorean Theorem). This tells us that the length (another word for absolute value) of the product of two complex numbers is the product of the lengths of the factors. Besides length, what else is needed to determine the line segment from O to the complex number z? One way to determine it is to use the angle from the positive x-axis to the segment from O to z. This angle is called the argument of z and is denoted . Note that arg(z) is determined only up to a multiple of radians and that it is not defined at all in case z = 0. In particular, we see that: , , , and (unless a = 0). Notice that our definition of the trigonometric functions using the unit circle automatically guarantees that all the signs in these formulas are correct regardless of quadrant in which the point z lies. In particular, we have which tells us that multiplying a complex number by a positive real number does not change the argument, but just expands the length by that factor. For example, doubling a complex number makes it twice as long but it points in the same direction. Let's calculate products using the argument function. Let have length and argument for j = 1, 2 (where we are assuming that neither is zero, since that case is trivial). The product is: where we the final simplification used the addition formulas for both the sine and the cosine function. So, the full story on multiplication is that you multiply the lengths of the factors and add the arguments: In the very special case of natural number powers this says: Proposition 16:(De Moivre) If a non-zero complex number z = a + bi has length r = \|z\| and argument and if n is any natural number, then: . Corollary 9: Let n be a natural number. There are (at least) n complex numbers which are -roots of 1. In fact, for k = 0, 1, ..., n - 1 all satisfy . More generally, every non-zero complex number z has (at least) n distinct roots. In fact, if z has length r and argument , then the following are all -roots of z: for all k = 0, 1, ..., n - 1. The corollary follows immediately from De Moivre's formulas. In fact, these are the only -roots, but it is convenient to defer the proof of this until a later chapter. The quantities of the first part of Corollary 9 are called -roots of unity. Geometrically, they all lie on the unit circle and are evenly spaced around the around the circle. The second part of the Corollary says that we can obtain the various roots of any number by simply multiplying any one of them by the various -roots of unity. A geometric figure is a collection of points. We can transform this set by applying operations to each of the points. For example, if you add a complex number to each point, it translates or shifts the figure. For example, if the figure is the unit circle, consisting of all the points (x,y) where . Then adding (2, 3) to each of these points gives the set of points (x + 2, y + 3) where . Letting x' = x + 2 and y' = y + 3, we see that x = x' - 2 and y = y' - 3. So, the shifted circle is the set of all (x', y') where . In general, let S be a set of points (x, y) where f(x, y) = 0. If we want to shift this a units to right and b units upward, then the new set of points is the set of (x, y) where f(x - a, y - b) = 0. For example, is a parabola with vertex at (2, 3). One can also do reflections across the y-axis by replacing x with -x. For example, the set of (x, y) where is the upper half of a parabola having the x-axis as its axis. Its reflection across the y-axis has equation . Similarly, one can reflect across the x-axis by replacing y with -y in the equation of the set. Note that this corresponds to mapping z = x + iy to its complex conjugate. The third type of transformation is a rotation about the origin. We know that we can rotate z = x + iy through an angle by multiplying it be the complex number to give the number zu = (xcos(theta) -ysin(theta)) + i(xsin(theta) + ycos(theta)). So, our new point is (x', y') where Alternatively, we can get x and y from x' and y' by rotating (x', y') through an angle . So, if the original set is the set of (x, y) where f(x, y) = 0, then the rotated set of (x, y) where . For example, suppose we want to rotate the hyperbola xy = 1 counter-clockwise 45 degrees. Then use and the equation of the rotated figure is or . Warning! It is notoriously easy to make a mistake in rotating in the wrong direction. You should always check a point afterwards to make sure you have rotated in the direction intended. The formula for doing the rotation is also hard to remember correctly; it is usually best to just remember that you rotate by multiplying by . When we set up the plane, we defined it to be the set of pairs (x, y) of real numbers. One then defined the distance between two points as by a formula involving the x and y coordinates of the two points. This means that our notion of distance appears to depend on the choice of coordinate system. In fact, it is independent of the choice of coordinate system as is straightforward to verify: Proposition 17: If two points and are transformed by a finite number or translations, rotations, and reflections, the distance between the two points does not change. In Chapter 1, we assumed an intuitive notion of what one meant by an angle -- it was measured as the length of the arc of the unit circle swept out as you traversed the angle. Although intuitive, it is difficult to define exactly what one means by the length of the arc of the circle swept out as you traverse the angle. In this section, we will see how to make this precise assuming that one has the basic properties of the sine and cosine function. In Chapter 5, we will complete the job by rigourously defining the trigonometric functions. First you need to remember that as ranges from 0 to radians, decreases from 1 to -1. So, for each real value v between -1 and 1, there is a unique such that cos(theta) = v. This uniquely defined is called the arccosine of v and is denoted either or . Next we need some definitions for complex numbers. If z = x + iy is a non-zero complex number, then the direction of z is defined to be u = z/\|z\|. (Geometrically, it is a complex number of length one pointed in the same direction as z.) For any complex number u = a + bi of length one, define For any non-zero complex number z, define arg(z) to be arg(u) where is the direction of z. The principal argument was defined so that In fact, it is the unique real number which satisfies this condition. It follows that if w and z are two non-zero complex numbers, then and where we say that two numbers are equal mod if their difference is an integer multiple of . Finally, in this section, we will refer to points (x, y) in the plane as if they are the corresponding complex number z = x + iy. A directed line segment AB is determined by its two endpoints A and B; so a directed line segment is really an ordered pair of points, which is the same thing as an ordered pair of complex numbers. Assuming that r and s are the complex numbers associated with A and B respectively, the complex number is called the direction of the directed line segment AB. Clearly the direction of AB is a complex number of length one, and the direction of BA is -u if u is the direction of AB. (Geometrically, u is a complex number of length one which points in the same direction as AB.) A directed angle is an ordered triple of points A, O, and B where both A and B are distinct from O. The measure of the directed angle is arg(u/v) where u is the direction of OA and v is the direction of OB. So, the measure of an angle is a real number in the interval such that . Note that the angles are directed in the sense that . One needs to be careful about this because many of the theorems in synthetic geometry refer to undirected angles, i.e. they use the absolute value of the measure of the angle so that for undirected angles. One can now prove many of the results of synthetic geometry. For example, Proposition 18: If is a triangle, then (as directed angles) one has Proof: Let the directions of BA, BC, and CA be u, v, and w respectively, Then the sum is the direction where the equality is mod . Now, if the sum of the three angles is not equal to , it must be equal to because each of the angles is in the interval . But this could only happen if all three angles were equal to , in which case would not be a triangle. Notice how easy it is to get the addition formulas: Proposition 19: (Addition Formulas) If A and B are two angles, then and Proof: Let u and v be the complex numbers of length 1 such that A = arg(u) and B = arg(v). Then A + B = arg(uv) (mod ) and -A = arg(1/u) = arg( ). One has and . Multiplying these expressions for u and v together givees the first two assertions. The last assertion follows from the definition of the complex conjugate. Proposition 20: (Law of Cosines) Let a, b, and c be the lengths of the sides of the triangle opposite angles A, B, and C respectively. Then Proof: Let u and v be the directions of CA and CB respectively. Then identifying the points with the corresponding complex numbers, we have B = C + av and A = C + bu. So, one has where one has used part iii of Proposition 19. Directions can be used to verify the usual properties of similar triangles. The exercises will give further details.
Functions and Their Graphs. Using Technology: Graphing a Function. The Algebra of Functions. Portfolio. Functions and Mathematical Models. Using Technology: Finding the Points of Intersection of Two Graphs and Modeling. Limits. Using Technology: Finding the Limit of a Function. One-Sided Limits and Continuity. Using Technology: Finding the Points of Discontinuity of a Function. The Derivative. Using Technology: Graphing a Function and Its Tangent Line. Summary of Principal Formulas and Terms. Review Exercises. 3. DIFFERENTIATION. Basic Rules of Differentiation. Using Technology: Finding the Rate of Change of a Function. The Product and Quotient Rules. Using Technology: The Product and Quotient Rules. The Chain Rule. Using Technology: Finding the Derivative of a Composite Function. Marginal Functions in Economics. Higher-Order Derivatives. Using Technology: Finding the Second Derivative of a Function at a Given Point. Implicit Differentiation and Related Rates. Differentials. Portfolio. Using Technology: Finding the Differential of a Function. Summary of Principal Formulas and Terms. Review Exercises. 4. APPLICATIONS OF THE DERIVATIVE. Applications of the First Derivative. Using Technology: Using the First Derivative to Analyze a Function. Applications of the Second Derivative. Using Technology: Finding the Inflection Points of a Function. Curve Sketching. Using Technology: Analyzing the Properties of a Function. Optimization I. Using Technology: Finding the Absolute Extrema of a Function. Optimization II. Summary of Principal Terms. Review Exercises. Antiderivatives and the Rules of Integration. Integration by Substitution. Area and the Definite Integral. The Fundamental Theorem of Calculus. Using Technology: Evaluating Definite Integrals. Evaluating Definite Integrals. Using Technology: Evaluating Definite Integrals for Piecewise-Defined Functions. Area between Two Curves. Using Technology: Finding the Area between Two Curves. Applications of the Definite Integral to Business and Economics. Using Technology: Consumers' Surplus and Producers' Surplus. Summary of Principal Formulas and Terms. Review Exercises. Functions of Several Variables. Partial Derivatives. Using Technology: Finding Partial Derivatives at a Given Point. Maxima and Minima of Functions of Several Variables. The Method of Least Squares. Using Technology: Finding an Equation of a Least-Squares Line. Constrained Maxima and Minima and the Method of Lagrange Multipliers. Double Integrals. Summary of Principal Terms. Review Exercises. Answers to Odd-Numbered Exercises. 0534419860 before 2pm cst. ...show less 05344198
0470500778 9780470500774 the Contents:Introduction How to use this book Overview of the examsPart I: Basic Skills Review Arithmetic and Data Analysis AlgebraPart II: Strategies and Practice Mathematical Ability Quantitative Comparison Data SufficiencyEach section includes a diagnostic test, explanations of rules, concepts withexamples, practice problems with complete explanations, a review test, and a glossary!Test-Prep Essentials from the Experts at CliffsNotes®For more test-prep help, visit CliffsNotes.com®*SAT is a registered trademark of the College Board, which was not involved in the production of, and does not endorse, this product. «Show less... Show more» Rent CliffsNotes Math Review for Standardized Tests 2nd Edition today, or search our site for other Kohn
Search Course Communities: Course Communities Dave's Short Trig Course A systematically presented multi-page course on basic trigonometry: trigonometry of triangles, laws of sines and cosines, angles, radian measure, trigonometric functions. The course includes an introduction that shows applications of trigonometry and provides motivation to study trigonometry. Presented as a sequence of linked HTML pages with embedded interactive Java applets.
Algebra for College Students 9780495105107 ISBN: 0495105104 Edition: 8 Pub Date: 2006 Publisher: Thomson Learning Summary: Kaufmann and Schwitters have built this text's reputation on clear and concise exposition, numerous examples, and plentiful problem sets. This traditional text consistently reinforces the following common thread: learn a skill; use the skill to help solve equations; and then apply what you have learned to solve application problems. This simple, straightforward approach has helped many students grasp and apply fundam...ental problem solving skills necessary for future mathematics courses in an easy-to-read format. The new Eighth Edition of ALGEBRA FOR COLLEGE STUDENTS includes new and updated problems, revised content based on reviewer feedback and a new function in iLrn. This enhanced iLrn homework functionality was designed specifically for Kaufmann/Schwitters' users. Textbook-specific practice problems have been added to iLrn to provide additional, algorithmically-generated practice problems, along with useful support and assistance to solve the problems for students
Elementary Linear Algebra - 4th edition Summary: The text starts off using vectors and the geometric approach, plus, it features a computational emphasis. The combination helps students grasp the concepts. At the same time, it provides a challenge for mathematics majors.BOOKWYNDER Pocasset, MA Portion of Proceeds Benefit the Big Brothers and Big Sisters of New England. Book is in "GOOD" Condition with has been previously read and unless stated in description has NO highlighting/underlinin...show moreg and is NOT a library book. Book will have visible wear to the cover
Standards for School Mathematics: Prekindergarten through Grade 12 What mathematical content and processes should students know and be able to use as they progress through school? Principles and Standards for School Mathematics presents NCTM's proposal for what should be valued in school mathematics education. Ambitious standards are required to achieve a society that has the capability to think and reason mathematically and a useful base of mathematical knowledge and skills. The ten Standards presented in this chapter describe a connected body of mathematical understandings and competencies—a comprehensive foundation recommended for all students, rather than a menu from which to make curricular choices. Standards are descriptions of what mathematics instruction should enable students to know and do. They specify the understanding, knowledge, and skills that students should acquire from prekindergarten through grade 12. The Content Standards—Number and Operations, Algebra, Geometry, Measurement, and Data Analysis and Probability—explicitly describe the content that students should learn. The Process Standards—Problem Solving, Reasoning and Proof, Communication, Connections, and Representation—highlight ways of acquiring and using content knowledge. » Growth across the Grades: Aiming for Focus and Coherence Each of these ten Standards applies across all grades, prekindergarten through grade 12. The set of Standards, which are discussed in detail in chapters 4 through 7, proposes the mathematics that all students should have the opportunity to learn. Each Standard comprises a small number of goals that apply across all grades—a commonality that promotes a focus on the growth in students' knowledge and sophistication as they progress through the curriculum. For each of the Content Standards, chapters 4 through 7 offer an additional set of expectations specific to each grade band. The Table of Standards and expectations in the appendix highlights the growth of expectations across the grades. It is not expected that every topic will be addressed each year. Rather, students will reach a certain depth of understanding of the concepts and acquire certain levels of fluency with the procedures by prescribed points in the curriculum, so further instruction can assume and build on this understanding and fluency. Even though each of these ten Standards applies to all grades, emphases will vary both within and between the grade bands. For instance, the emphasis on number is greatest in prekindergarten through grade 2, and by grades 9–12, number receives less instructional attention. And the total time for mathematical instruction will be divided differently according to particular needs in each grade band—for example, in the middle grades, the majority of instructional time would address algebra and geometry. Figure 3.1 shows roughly how the Content Standards might receive different emphases across the grade bands. Fig. 3.1. The Content Standards should receive different emphases across the grade bands. p. 30 This set of ten Standards does not neatly separate the school mathematics curriculum into nonintersecting subsets. Because mathematics as a discipline is highly interconnected, the areas described by the Standards overlap and are integrated. Processes can be learned within the Content Standards, and content can be learned within the Process » Standards. Rich connections and intersections abound. Number, for example, pervades all areas of mathematics. Some topics in data analysis could be characterized as part of measurement. Patterns and functions appear throughout geometry. The processes of reasoning, proving, problem solving, and representing are used in all content areas. The arrangement of the curriculum into these Standards is proposed as one coherent organization of significant mathematical content and processes. Those who design curriculum frameworks, assessments, instructional materials, and classroom instruction based on Principles and Standards will need to make their own decisions about emphasis and order; other labels and arrangements are certainly possible. Where Is Discrete Mathematics? The 1989 Curriculum and Evaluation Standards for School Mathematics introduced a Discrete Mathematics Standard in grades 9–12. In Principles and Standards, the main topics of discrete mathematics are included, but they are distributed across the Standards, instead of receiving separate treatment, and they span the years from prekindergarten through grade 12. As an active branch of contemporary mathematics that is widely used in business and industry, discrete mathematics should be an integral part of the school mathematics curriculum, and these topics naturally occur throughout the other strands of mathematics. p. 31 Three important areas of discrete mathematics are integrated within these Standards: combinatorics, iteration and recursion, and vertex-edge graphs. These ideas can be systematically developed from prekindergarten through grade 12. In addition, matrices should be addressed in grades 9–12. Combinatorics is the mathematics of systematic counting. Iteration and recursion are used to model sequential, step-by-step change. Vertex-edge graphs are used to model and solve problems involving paths, networks, and relationships among a finite number of objects. »
Wednesday, January 12, 2011 Virtual Nerd Review As part of my job as a reviewer for The Old Schoolhouse Review Crew, I recently reviewed Virtual Nerd, which is a website designed to help students in Pre-Algebra, Algebra I, Algebra II, and Physics. Virtual Nerd offers a tutoring option to students who need help in those subjects other than a private tutor. According to Virtual Nerd: Virtual Nerd offers online tutoring in math and science to grades 7-12. Our service is both affordable and convenient, providing help whenever and as much as needed. Using our patent pending Dynamic Whiteboard, each student has a very interactive and personalized learning experience. While watching our pre-recorded videos, they can drill-down for more information and access FAQ's terms and definitions to assist in their learning process. Service is available on a subscription plan basis, a month subscription is $49 - multi-month plans are also available. Before going any further with my review, I would encourage you to check Virtual Nerd out for yourself. You can try all Virtual Nerd has to offer free of charge for 2 hours. As well, you can try Virtual Nerd for just one day (one time only) for $5.00 or for a week (one time only) for $19.00. As you can see, Virtual Nerd offers several inexpensive options with which you can explore the website and see if it would work for your family. For the purposes of this review, my daughter and I used the site to access some of the video tutorials for Algebra I as this is the math course she is taking right now. From the Algebra I Search page, one can search by: Keyword Topic Textbook Search (Virtual Nerd currently corresponds with Prentice Hall Algebra 1, 2004; Holt Algebra 1, 2007; Glencoe Algebra 1, 2004; and McDougal Littell Algebra 1, 2007). Right now, only Algebra has the option of the textbook search in which specific tutorials are linked to chapter and lesson in the textbook. Major topics included in the Algebra 1 section include: getting ready for algebra foundations for algebra solving linear equations relations and functions analyzing linear equations solving and graphing linear inequalities systems of equations and inequalities exponents and exponential functions polynomials and factoring quadratic equations and functions radical expressions and equations rational expressions and functions probability and data analysis. My daughter and I found the video tutorials very clear and easy to follow. The video tutorials consisted of a real person talking very clearly and writing on a white board. At the end of the video, other videos on the same topic are suggested so that the student can easily link to other, related tutorials that will be helpful. As well, the student can watch the same tutorial again very easily. The only problem I had with Virtual Nerd was that, when we were first using the site, the tutorials would not load for viewing on our computer. After updating our internet browser, though, I no longer had any problems with the site. The amazingly quick customer service I received in response to my problem was very much appreciated. Even though I e-mailed my question late in the evening, I received a response within 30 minutes! While Virtual Nerd currently offers over a thousand different tutorials, they plan to expand their tutorials to cover even more Physics, Chemistry, Pre-Calculus, and Calculus in the 2011-2012 school year. If you have a child struggling in Pre-Algebra, Algebra I, Algebra II, or Physics, Virtual Nerd offers the supplemental teaching help that may be a perfect fit for your family. I received a temporary subscription to Virual Nerd in exchange for this review. No other compensation, monetary or otherwise, was given in exchange for this review. 2 comments: I thought you would like to know that because Virtual Nerd ( has received such positive feedback and significant interest following your review of their online tutoring service, they are offering a special discount to homeschool parents. Homeschool parents can receive 50% off the 1 and 3 month subscription plans. Just enter the discount code: homeschooldeal when you sign up for service. That is a huge discount, and hurry - the offer is valid until March 31, 2011
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education. Managing the effect of incidental take on the population viability of endangered species A dissertation research proposal by Conor McGowan Objective 1: Modeling the effect of incidental take of Piping Plover eggs and chicks on population viability i APPLICATION FORM 1) Name (last name, first name, middle initial)_ 2) Country of citizenship and passport number_ 3) Present address__ 4) Telephone with area code_ Above address effective through what date?_ 5) Address and telephone after end of schoo TAKE HOME QUIZ #8:NAME: . ID#: . SECTION: .SHOW YOUR WORK. NO WORK, NO CREDIT. IN THE FOLLOWING PROBLEMS, YOU MAY USE EQUA\|SOLV TO SOLVE ANY EQUATION. HOWEVER, YOU MUST STATE THE EQUATION THAT YOU ARE ABOUT TO SOLVE AND INDICATE THAT YOU WILL USE Success in MathematicsUniversity of Missouri Department of Mathematics Fall Semester 2008The following are some of my ideas about how to do well and truly obtain something out of a mathematics course. I have listed tips on how to study mathematics Teaching StatementUniversity of Missouri Michael Pemberton Graduate Teaching AssistantEvery teacher has his or her beliefs about teaching. My beliefs are based upon the non-authoritarian mindset, more specifically progressivism. I believe students Success in MathematicsUniversity of Missouri Department of Mathematics Winter Semester 2009The following are some of my ideas about how to do well and truly obtain something out of a mathematics course. I have listed tips on how to study mathemati Success in MathematicsUniversity of Missouri Department of Mathematics Fall Semester 2006The following are some of my ideas about how to do well and truly obtain something out of a mathematics course. I have listed tips on how to study mathematics Success in MathematicsUniversity of Missouri Department of Mathematics Summer Semester 2008The following are some of my ideas about how to do well and truly obtain something out of a mathematics course. I have listed tips on how to study mathemati MATH 1100 (5-DAY) COURSE SYLLABUS - FS2006Your Homework:You are responsible to know how to do all the problems assigned on the LAB WORKSHEETS, all the PRACTICE EXERCISES, and all the TEXTBOOK'S worked out examples as your exam problems will be simi Success in MathematicsUniversity of Missouri Department of Mathematics Winter Semester 2008The following are some of my ideas about how to do well and truly obtain something out of a mathematics course. I have listed tips on how to study mathemati Math 1300 Name:QUIZ #3Date: . .. ID # . Section: You must show all your work to receive full credit!1.(5 Points)Six houses in a row are each to be painted with one of the colors red, blue, green, and yellow. In how many dierent ways can t Math 1300 Name:QUIZ #1Date: . .. ID # . Section: You must show all your work to receive full credit!1.(10 Points)On December 31, 1995, a house is purchased with the buyer taking out a 30-year $100, 000 mortgage at 9% interest compounded m Math 1300 Name:QUIZ #7Date:. .. ID # . Section: You must show all your work to receive full credit!1.(10 Points)Statistics show that 61% of all married women in the United States are in the labor force. Eight married U.S. women are rand Math 1300 Name:QUIZ #6Date:. .. ID # . Section: You must show all your work to receive full credit!1.(5 Points)At a local college, 4 sections of economics are taught during the day and 2 sections are taught at night. 75% of the day sect Math 1300 Name:QUIZ #4Date:. .. ID # . Section: You must show all your work to receive full credit!1.(5 Points)Which of the following is a valid probability distribution for a sample space containing outcomes r, s, t, v. (a) (b) (c) (d) Math 1300 Name:QUIZ #8Date: . .. ID # . Section: You must show all your work to receive full credit!1.(10 Points)Find the equation of the line through (1, 1) and the intersection point of the lines 3x + 3y = 3 and 8x 4y = 12..A Typese Math 1300 Name:QUIZ #2Date: . .. ID # . Section: You must show all your work to receive full credit!1.(10 Points)Let R, S, and T be subsets of the universal set U. Draw a Venn diagram and use the data given below to label the number of el Math 1100 Name:QUIZ #15Date: . .. ID # . Section: You must show all your work to receive full credit!1.(5 Points)Solve and graph the solution set form 6<3m2 3< 7.. 2. (5 Points) Solve and graph the solution set for 2x+1 > 5. 3 W Math 0110Quiz 121Name: . ID #: . Section: .Answer all questions and show all of your work. Answers which are not justified with appropriate work will receive 0 points. Calculators are not permitted! 5 1. Perform the indicated operations to si Math 0110Quiz 101Name: . ID #: . Section: .Answer all questions and show all of your work. Answers which are not justified with appropriate work will receive 0 points. Calculators are not permitted! 4 1. Determine the domain of the rational e 91Name: . ID #: . Section: .Answer all questions and show all of your work. Answers which are not justified with appropriate work will receive 0 points. Calculators are not permitted! 1. Solve each of the following equations. 5 141Name: . ID #: . Section: .Answer all questions and show all of your work. Answers which are not justied with appropriate work will receive 0 points. Calculators are not permitted! 10 1. Solve the following rational equation 0110Quiz 101Name: . ID #: . Section: .Answer all questions and show all of your work. Answers which are not justied with appropriate work will receive 0 points. Calculators are not permitted! 4 1. Determine the domain of the rational exp EE/CS 6760 Parallel Computer ArchitectureMidterm ExamThis take-home midterm is being given on the honor system. You have 48 hours to complete this exam once you begin. You are to work individually. You are free to consult Hwang, any other texts, or Math 0110Quiz 111Name: . ID #: . Section: .Answer all questions and show all of your work. Answers which are not justified with appropriate work will receive 0 points. Calculators are not permitted! 5 1. Solve the following rational equation. 71Name: . ID #: . Section: .Answer all questions and show all of your work. Answers which are not justified with appropriate work will receive 0 points. Calculators are not permitted! 3 1. Use scientific notation to simplify, th Math 0110Quiz 21Name: . ID #: . Section: .Answer all questions and show all of your work. Answers which are not justified with appropriate work will receive 0 points. Calculators are not permitted! 6 1. Simplify the expression. 15 - {12 - (10 Math 0110Quiz 61Name: . ID #: . Section: .Answer all questions and show all of your work. Answers which are not justified with appropriate work will receive 0 points. Calculators are not permitted! 1. The value of a computer bought in the yea Math 0110Quiz 141Name: . ID #: . Section: .Answer all questions and show all of your work. Answers which are not justified with appropriate work will receive 0 points. Calculators are not permitted! 10 1. Solve the following rational equation Math 0110Quiz 61Name: . ID #: . Section: .Answer all questions and show all of your work. Answers which are not justied with appropriate work will receive 0 points. Calculators are not permitted! 1. The value of a computer bought in the year Math 0110Quiz 11Name: . ID #: . Section: .Answer all questions and show all of your work. Answers which are not justied with appropriate work will receive 0 points. Calculators are not permitted! 6 1. Determine the gcd and lcm of 180 and 480 1100 Name:QUIZ #17Date:.. ID # . Section: You must show all your work to receive full credit!.1.(5 Points)Let f (x) =3x2 -12x+10 . x2 -4x+4Use this function to answer the following questions.(a) Use long division to express Math 1100 Name:QUIZ #8Date:. .. ID # . Section: You must show all your work to receive full credit!1.(5 Points)Find the equation of the perpendicular bisector, L, of AB, if A(2, 7) and B(8, 9). Label your work and tell what you are ndin MATH 1100: TAKE HOME QUIZ #2Due in Class on Monday, October 2, 2006 LAST NAME: FIRST NAME. ID#: . SECTION #: . INSTRUCTORS NAME: Please print this sheet, and use it to work the problems. Have all pages stapled together before handing in your work Math 1100 Name:QUIZ #6Date:. .. ID # . Section: You must show all your work to receive full credit!1.(10 Points)Write an equation for a circle satisfying the given conditions in standard and general forms. The points (7, 13) and (3, 11) Math 1100 Name:QUIZ #18Date:.. ID # . Section: You must show all your work to receive full credit!.1.(5 Points)Use f (x) =3x2 5x+4to complete the following parts.(a) Show algebraically that the function f (x) is one-to-one. (b) Math 1100 Name:QUIZ #14Date: . .. ID # . Section: You must show all your work to receive full credit!1.(3 Points)Solve the following expression for x.9 x2 +3x+9=x x-3-27 x3 -27. 2. (4 Points) Solve the following expression for Math 1100 Name:QUIZ #1Date: . .. ID # . Section: You must show all your work to receive full credit!1.(3 Points)The area of Hong Kong is 412 square miles. It is estimated that the population of Hong Kong will be 9,600,000 in 2050. Find th Math 1100 Name:QUIZ #9Date: . .. ID # . Section: You must show all your work to receive full credit!1.(5 Points)A rectangular box with volume 108 m3 is built with a square base and top. The cost is $0.90/m2 for the bottom, $3.75/m2 for th MATH 1100: TAKE HOME QUIZ #1LAST NAME: . FIRST NAME. ID#: . SECTION #: . INSTRUCTORS NAME: . Please print these sheets, and use them to work the problems. Have all pages stapled together. Show all your work in detail. Just an answer without work sh
9780314069Thomson Advantage Books: Algebra and Trigonometry In this new ADVANTAGE SERIES version of David Cohen's ALGEBRA AND TRIGONOMETRY, Fourth Edition, Cohen continues to use the right triangle approach to college algebra. A graphical perspective, with graphs and coordinates developed in Chapter 2, gives students a visual understanding of concepts. The text may be used with any graphing utility, or with none at all, with equal ease. Modeling provides students with real-world connections to the problems. Some exercises use real data from the fields of biology, demographics, economics, and ecology. The author is known for his clear writing style and numerous quality exercises and applications. As part of the ADVANTAGE SERIES, this new version will offer all the quality content you've come to expect from Cohen sold to your students at a significantly lower price
Brief Calculus And Its Applications - 8th edition Summary: Once again, these extremely readable, highly regarded, and widely adopted texts present "tried and true" formula pairing substantial ...show moreamounts of graphical analysis and informal geometric proofs with an abundance of hands-on exercises has proven to be tremendously successful with both students and instructors. What would the benefit to your students be of using a text which blends practical applications with mathematical concepts? Features NEW - Details the ways in which technology can be used to foster understanding of several topics while it facilitates computation. NEW - Ends each chapter with a Review of Fundamental Concepts, helping students focus on the chapter's key points. NEW - Places greater emphasis on the significance of differential equations in applications involving exponential functions. NEW - Customized calculus software is available through the study guide. NEW - Companion website supports and extends the materials presented in the text. NEW - All graphs of functions have been redrawn using Mathematicia. Reinforces class lessons with carefully designed exercise sets, and challenges students to make their own connections. Minimizes prerequisites, allowing those who have forgotten much of their high school mathematics to start anew with this self-contained material. Functions and Their Graphs. Some Important Functions. The Algebra of Functions. Zeros of Functions. The Quadratic Formula and Factoring. Exponents and Power Functions. Functions and Graphs in Applications. 1. The Derivative The Slope of a Straight Line. The Slope of a Curve at a Point. The Derivative. Limits and the Derivative. Differentiability and Continuity. Some Rules for Differentiation. More About Derivatives. The Derivative as a Rate of Change. 2. Applications of the Derivative Describing Graphs of Functions. The First and Second Derivative Rules. Curve Sketching (Introduction.) Curve Sketching (Conclusion.) Optimization Problems. Further Optimization Problems. Applications of Calculus to Business and Economics. 3. Techniques of Differentiation The Product and Quotient Rules. The Chain Rule and the General Power Rule. Implicit Differentiation and Related Rates. Antidifferentiation. Areas and Reimann Sums. Definite Integrals and the Fundamental Theorem. Areas in the xy-Plane. Applications of the Definite Integral. 7. Functions of Several Variables Examples of Functions of Several Variables. Partial Derivatives. Maxima and Minima of Functions of Several Variables. Lagrange Multipliers and Constrained Optimization. The Method of Least Squares. Double Integrals. 8. The Trigonometric Functions Radian Measure of Angles. The Sine and the Cosine. Differentiation of sin t and cos t. The Tangent and Other Trigonometric Functions. 9. Techniques of Integration Integration by Substitution. Integration by Parts. Evaluation of Definite Integrals. Approximation of Definite Integrals. Some Applications of the Integral. Improper Integrals
Differential Equations Spring 2008 Homework Write-Ups Problem write-ups are your permanent record of your understanding of the material covered. This is especially true in a course such as this where there are no exams. Solutions should be clearly and logically presented. This means that: Your method should always be clear. It should be easy to figure out what you're doing and why. Use a lot of space. I recommend skipping some lines if you use lined paper. Equations should usually be accompanied by prose. Before plunging into algebra, state what it is you're solving for. If there are any non-obvious steps in a calculation, explain them. Write equations in a logical order. Most of the problems in this course are not short plug-ins. They will require you to work through a multiple-step process, often devising and testing a mathematical model along the way. It is absolutely essential in such problems that you explain your reasoning clearly. For these sort of problems, the explanation and narrative is the solution. Solutions should stand on their own; they should be understandable to someone who hasn't read the problem. This means that you should paraphrase the question before writing your response. For many problems you will find yourself using Maple. For all but the simplest Maple calculations you should include a printout of your Maple worksheet. I will not give numerical grades on HW assignments. Instead, I will give a letter grade and try to include as many comments as I can. I'm mainly interested in seeing that you thoughtfully attacked the problem and wrote it up in a clear and coherent way. Finally, a few minor requests: On the top of the homework, please write the assignment number. If you don't have a stapler, that's ok. But please don't mangle and fold over the corner in an attempt to get the pages to stick together. Just write your name or initials on all pages and I'll gladly staple them together. Please don't hand in problems on paper that has been torn out of a spiral notebook.
Math Errors - Learn From Them "The most powerful learning experiences often result from making mistakes". I usually address my students with the above phrase after handing out marked papers, tests and exams. I then provide time for my students to carefully analyze their errors. I also ask them to keep a running record/journal of the patterns of their errors. Understanding how and where you go wrong will lead to enhanced learning and improved grades - a habit often developed by strong math students. It's not unlike me to develop my next test based on a variety of student errors! How often have you looked over your marked paper and analyzed your errors? When doing so, how many times have you almost immediately realized exactly where you went wrong and wished that if only you had caught that error prior to submitting your paper to your instructor? Or, if not, how often have you looked closely to see where you went wrong and worked on the problem for the correct solution only to have one of those 'A Ha' moments? 'A Ha' moments or the sudden enlightening moment resulting from the newly discovered understanding of the misconceived error usually means a breakthrough in learning, which often means that you'll rarely repeat that error again. Instructors of mathematics often look for those moments when they are teaching new concepts in mathematics; those moments result in success. Success from previous errors isn't usually due to the memorization of a rule or pattern or formula, rather, it stems from a deeper understanding of 'why' instead of 'how' the problem was resolved. When we understand the 'whys' behind a mathematical concept rather than the 'hows', we often have a better and deeper understanding of the specific concept. Symptoms & Underlying Causes of Errors When reviewing the errors on your papers, it's crucial that you understand the nature of the errors and why you made it (them). I've listed a few things to look for: Application errors (misunderstanding of one or more of the required step(s) Knowledge based errors (lack of knowledge of the concept, unfamiliar with terminology) Order of Operations (often stems from rote learning as opposed to having a true understanding) Incomplete (practice, practice and practice, this leads to having the knowledge more readily available) Success is Failure Inside Out! Think like a mathematician and learn from your previous mistakes. In order to do so, I would suggest that you keep a record or journal of the patterns of errors. Mathematics requires a lot of practice, review the concepts that caused you grief from previous tests. Keep all of your marked test papers, this will assist you to prepare for ongoing summative tests. Diagnose problems immediately! When you are struggling with a specific concept, don't wait to get assistance (that's like going to the doctor three days after breaking your arm) get immediate help when you need it, if your tutor or instructor isn't available - take the initiative and go online, post to forums or look for interactive tutorials to guide you through.
You are here Mathematics of Musical Signals 2: The Wave Equation Department - LART Offered - Spring Course number - LMSC-P315 In this course, students explore the ways that symbolizing musical signals contributes to the design and development of sound. Students study the mathematics behind acoustic and electrical signals. This course continues the exploration of the mathematics behind musical signals that began in LMSC-P310. Students use mathematics to analyze musical signals. They evaluate complex waveforms using mathematics. And they apply mathematics to signals to understand transformation. Students explore resonance and the wave equation. In addition, students learn further how to describe and manipulate mathematically musical signals and their representations.
This is the ultimate mathematics course. We will cover everything from Algebra I and work our way up to Calculus concepts with some great discussions along the way. Mathematics is a language and if you can become fluent in it (which if you complete this course, I promise you will be!), then that is simply one more step in your great knowledge bank. (Every student has the option to pick a certain semester or set of semesters to learn from, as I do know that this is a very long course. Thus, if there is simply a few things that you would like to work on, pick a semester and let me know. If you just want to get the whole feel of the course, I will be glad to start from the beginning.) Please know that I am a bit more of a lecture type of teacher, and thus, you may hear me speak quite a lot. However, within mathematics, the main thing we try to accomplish is your knowledge of the basic priniciples, and thus, mathematics consists of repetition, repetition, repetition! Here is the basic outline of a daily lesson: Get acquainted and review material from last lesson Teach new material Practice new material Q/A Session Assigned Work Now, you will most likely learn a few new principles in one lesson, so steps 2 and 3 may be repeated a few times. Homework will be given daily, and you can usually just download the homework from clicking the links below. Tests will be at least once a week, and unit tests will be about every 3 - 4 weeks, unless changed by me. Attendance is crucial in this type of course! If you are not here for two to three days at a time, then we will not be able to effectively absorb the knowledge that is being presented to you, and thus, it will take longer to learn this. At the end of every semester, you will have a final exam which will cover all of what you have learned. Below lists all semesters and lessons that will be involved with your course. Next to each lesson is a set number of problems. This is basically how much homework you will have for each lesson, and you will have until the end of the week to get all problems completed. Soon, each set of problems will have a hyperlink within them that goes to a page where the assignment is, and you can simply do the problems from there. These are listed so that potential students can examine how much work will be involved in each semester, and how you can estimate the approximate time that you will need to devote to this course.
International Society for Bayesian Analysis (ISBA) Promotes the development of Bayesian statistical theory and its application to problems in science, industry and government. News, history and minutes, archive of abstracts, information about the Reverend Thomas Bayes, and open positions in the field.Japanese Association of Mathematical Sciences (JAMS) A scientific research organization whose main activity is to publish scientific journals in English, French or German, in particular Scientiae Mathematicae Japonicae. Journal and submission information, newsletter, and meetings. Also available in JapaneseMarc Chamberland A mathematician at Grinnell College interested in differential equations and dynamical systems. Resources for the 3x + 1 problem and the Jacobian Conjecture include papers to download in PostScript format and information and proceedings for related conferences. ...more>> Mathematics of Planet Earth Mathematics of Planet Earth (MPE) "provides a platform to showcase the essential relevance of mathematics to planetary problems, coalesces activities currently dispersed among institutions, and creates a context for mathematical and interdisciplinary ...more>> Minnesota Council of Teachers of Mathematics (MCTM) An affiliate of the National Council of Teachers of Mathematics (NCTM). The site provides information about conferences, events, and programs; membership; listings of recommended math sites; MCTM highlights (Presidential Awardees, Shape of Space video, ...more>> MOTIVATE - Univ. of Cambridge, UK MOTIVATE is a project incorporating a series of videoconferences, run by the Millennium Mathematics Project at Cambridge. The objectives of MOTIVATE are: to enrich the mathematical experience of school students, to broaden their mathematical horizons, ...more>> MSPnet: The Math and Science Partnership Network - TERC The Math and Science Partnership (MSP) Program is a major research and development effort to understand and improve the performance of K-12 students in mathematics and science. MSPnet is their electronic community. Learn about individual partnership projects; ...more>> National Council of Supervisors of Mathematics (NCSM) A resource site for those interested in leadership in mathematics education. The site lists meetings and conferences, membership information and how to subscribe to the NCSM mailing list, publications, operations, information about the summer Leadership ...more>> SyllabusWeb - Syllabus Press, Inc. From the publishers of Syllabus Magazine, a technology magazine for high schools, colleges, and universities. Highlights of recent issues of the magazine and full text archives of all Press publications. The June 1995 issue covers telecommunications and ...more>>
Basic Skills 7: In this course, students review basic math operation, number sense, functions and applications, measurements, geometry, and algebra. _____________________________________________________________________ Student Testimonies The College Readiness Academy Program has helped me accomplish my educational goals. The free tutoring is helpful because it assists me with my class work and homework. The teachers inform us about other college services that are avaliable for us. ~ Griselda Espinoza This program is great because it motivates and helps me focus on what I want for my future.
... read more Customers who bought this book also bought: Our Editors also recommend: Mathematician's Delight by W. W. Sawyer "Recommended with confidence" by The Times Literary Supplement, this lively survey was written by a renowned teacher. It starts with arithmetic and algebra, gradually proceeding to trigonometry and calculus. 1943The Nature of Mathematics by Philip E. B. Jourdain Anyone interested in mathematics will appreciate this survey, which explores the distinction between the body of knowledge known as mathematics and the methods used in its discovery. 1913 editionProduct Description: idean geometry, matrices, determinants, group theory, and related topics. 1955
In our world ofgrowing technology, it is essential that students learn to work with and adapt to a variety of tools. The LCHS Math Department regularly utilizes graphing calculator technology to enhance lessons and provide more access to advanced mathematical thinking. We highly recommend that each student have his or her own graphing calculator for use during and outside of class, when deemed appropriate by his/her math teacher. Since we feel it is important for students to develop strong mental math and reasoning skills, there will be times when students are not permitted to use acalculator. Our teachers usethe TI-83, TI-84 Plus, TI-84 Plus Silver Edition or the TI-Nspire (with TI-84 face plate) calculators. We recommend that students use one of these calculators so that our teachers will be able to help quickly trouble-shoot problems and provide consistent instruction on using the graphing calculator technology. All of these calculators are permitted on most standardized tests including theSATs, SOLs, and AP Exams. Students taking AP Calculus AB, BC, or AP Statistics may want to purchase the TI-89 Titanium orthe TI-Nspire CAS because these calculators are equipped to deal with the advanced topics in these courses. To find more information about these calculators, you may want to visit education.ti.com andclick on the "Products" tab. These calculators are available at most electronics stores, office supply stores, or can be found on eBay. A limited number ofcalculators will be available for students who are financially unable to purchase their own calculator. There will be a $120 replacement fee charged to students if a calculator is lost or damaged. Forms will be available through the math teachers during the first week of school for students wishing to check-out a calculator to use for the school year. Calculators will be issued for the 2012-2013 school year on a first-come, first-served basis starting at 3:50 pm on Thursday, August 30 in the math wing hallway. If you have anyquestions about getting a calculator for your student, please contact his orher math teacher or the Math Department Chair, Nicole Kezmarsky (Nicole.Kezmarsky@lcps.org). HelpUs Get Cool Calculator Accessories with TI Points!! Did you just purchase a new TI Calculator? If you look on the back of the package, you willsee a TI Rewards symbol: Bring this in toyour math teacher so that we can collect point and use them to purchase greataccessories to use in class!!
Buy PDF List price: $30.00 Our price: $27.99 You save: $2.01 (7%) Standards-Driven Power Algebra I is a textbook and classroom supplement for students, parents, teachers and administrators who need to perform in a standards-based environment. This book is from the official Standards-Driven Series (Standards-Driven and Power Algebra I are trademarks of Nathaniel Max Rock). The book features 412 pages of hands-on standards-driven study guide material on how to understand and retain Algebra I. Standards-Driven means that the book takes a standard-by-standard approach to curriculum. Each of the 25 Algebra I standards are covered one-at-a-time. Full explanations with step-by-step instructions are provided. Worksheets for each standard are provided with explanations. 25-question multiple choice quizzes are provided for each standard. Seven, full-length, 100 problem comprehensive final exams are included with answer keys. Newly revised and classroom tested. Author Nathaniel Max Rock is an engineer by training with a Masters Degree in business. He brings years of life-learning and math-learning experiences to this work which is used as a supplemental text in his high school Algebra I classes. If you are struggling in a "standards-based" Algebra I class, then you need this book! (E-Book ISBN#0-9749392-1-8 (ISBN13#978-0-9749392-1-6)) (Perfect Bound Book ISBN#0-9749392-0-x (ISBN13#978-0-9749392-0-9))
Mathematics This first year regents course studies many major topics in mathematics including solving equations, factoring polynomials, probability, statistics, graphing functions, number systems, and real world applications. Integrated Algebra Extended 1 is the first year of a two year extended regents course. This course will study the same content as the Integrated Algebra Regents course, but will provide more time to develop specific algebra skills. Integrated Algebra Extended 2 is the second year of a two year extended regents course. This course will study the same content as the Integrated Algebra Regents course, but will provide more time to develop specific algebra skills. Pre-Calculus is designed to provide students with a solid foundation for calculus. This course provides an extensive treatment of the necessary topics from algebra, trigonometry, and analytic geometry. The objective of Financial Topics 1 is to provide students with a solid foundation for the application of high school mathematics in the real world. The focus of the course will be to prepare students for the reality of finances in the real world. The objective of Financial Topics 2 is to continue to provide students with a solid foundation of applications of mathematics in the real world. Topics will include mathematical appreciation, design and engineering mathematics, real estate, retirement planning and building a house. Computer Programming 1 is an introduction to the problem solving tools needed to successfully design, write, and debug computer programs using Visual Basic. The goal of this course is to introduce Visual Basic and its wide variety of tools to students. Computer Programming 2 is an introduction to the problem solving tools needed to successfully design, write, and debug computer programs using Visual Basic. The goal of this course is to introduce Visual Basic and its wide variety of tools to students.
Email This Article You successfully shared this article! Email This Article How to Do Algebra in Your Head There is a free online book called Inner Algebra provides ways and tricks to solve algebra in your head. This is extremely useful for student (and people who want to impress people). Methods include visualization, chunking, windowing , correspondence. Most of the techniques involve practice your mind to think on steps visually: … There are a few key abilities that form a foundation for doing math intuitively. The first is visualization. It is similar to using your imagination. If you have always been a 'visual' person, you will find some things easier. The first part of this chapter helps you develop this ability. If you already feel skilled in this area, this section can help you strengthen it in the specific ways that help you use it for math. Some people are naturally quite good at visualizing. If you feel that you don't really need training in this area, you can skip the first section. Just make sure you can do the exercises at the end of the chapter…. it may be true that its written by an amature and it would be wrong to represent it otherwise – i think this novell approach is very valuable and learning to approach math in various ways is extremely useful and important, if you read stories of savants – you will learn that they do math in their mind "visually" the visualize very clear images often very complex and are able to do computations that would be very complex even for those proficient in math – so the more we can train our mind the better. even if methods suggested are incomplete
MATH 090 Basic Math 5 Credits Operations with whole numbers, fractions, decimals and signed integers. Also includes percents, proportions, measurement, basic geometry, applications, problem-solving strategies, and writing about mathematics. Prerequisite: Appropriate placement score on the Arithmetic Test or recommendation from ABE. Back To Top MATH 094 Basic Algebra5 Credits An introduction to algebra including operations with signed numbers, solving basic linear equations, graphing linear relationships, problem solving, positive properties of exponents, and addition and subtraction of polynomials. Prerequisite: MATH 090 with a "C" or better or appropriate placement test score required. Back To Top MATH 098 Elementary Algebra5 Credits Review of operations with signed numbers and solving basic linear equations. Graphing linear relationships, the equation of a line, systems of equations, properties of exponents, operations on polynomials, and solving quadratic equations by factoring. Prerequisite: MATH 090 with a "C" or better or MATH 094 or appropriate placement test score required. Back To Top MATH 099 Intermediate Algebra5 Credits Simplifying and solving rational, radical and quadratic expressions and equations. Linear, quadratic, and exponential, functions with their graphs and applications. Prerequisite: MATH 096 or MATH 098 with a grade of "C" or better or appropriate placement test score required. Back To Top MATH 100 Mathematics for Early Childhood Educators5 Credits A course for early childhood educators focusing on math concepts appropriate for young children. Topics include patterns, sequencing, classifying, number systems and computation, functions, geometry, measurement, and basic concepts from statistics and probability. Interactive, activity-based methods are used guided by national and state mathematics education standards. Emphasizes conceptual understanding, connections among topics, and communication of mathematical thinking. Prerequisite: MATH 094 or appropriate placement test score. Back To Top MATH 101 Technical Mathematics I5 Credits Theory and applications of mathematics used in technical fields with emphasis on problem solving strategies, measurement, algebra, geometry, unit conversions and the metric system. Prerequisite: MATH 094 with a miminum grade of "C" or MATH 098 with a minimum grade of "C" or appropriate placement test score. Back To Top MATH 102 Technical Mathematics II5 Credits Emphasis on right triangle trigonometry and oblique triangle applications involving the Law of Sines, Law of Cosines, and vectors. Algebraic concepts, such as Pythagorean Theorem and vectors, integrated with applications of geometry, trigonometry, and physics. Formulas for area, perimeter, and volume are applied to composite shapes and optimization problems. Prerequisite: MATH 098 (Grade "C" or better). Back To Top MATH& 107 Math in Society 5 Credits An introduction to methods of thought in mathematics. Surveys the history of mathematics to reveal the multi-cultural and international nature of mathematics. Other topics chosen from: Problem-solving strategies, logic, sets, number theory, geometry, probability and statistics, functions and graphs, axiomatic systems. This course was formerly known as MATH 107, Math for LIberal Arts. Prerequisite: MATH 099. Back To Top MATH& 131 Math for Elementary Education I 5 Credits For prospective or practicing elementary teachers focusing on the mathematics underlying modern elementary school math. Topics include: number systems, models for operations, problem-solving techniques, algebraic thinking, appropriate technology and a variety of instructional approaches. Emphasizes deep conceptual understanding of content, connections among topics, communication of mathematical ideas and the developmental progression of topics. Prerequisite: MATH 099 or appropriate math placement test score and eligible for ENGL& 101 or WRIT 101. Back To Top MATH& 132 Math for Elementary Education II 5 Credits Delves deeply into the mathematics elementary teachers are responsible for teaching at the K-8 levels in the areas of geometry, measurement and probability. Emphasizes deep conceptual understanding of content, multiple representations, and communication of mathematical ideas. Appropriate technology is incorporated. Recommended for prospective and practicing elementary school teachers. Prerequisite: MATH& 131 (2.0 or better). Back To Top MATH& 141 Precalculus I 5 Credits Elementary functions with an emphasis on polynomial functions, rational functions, exponential functions and logarithmic functions. This course was formerly known as MATH 121. Prerequisite: A grade of "C" or better in MATH 099 or (MATH 120 or MATH 140 now retired) or appropriate test scores. Back To Top MATH& 142 Precalculus II 5 Credits Elementary functions with an emphasis on trigonometric functions and their applications, analytic geometry and polar cooridinates. This course was formerly known as MATH 122. Prerequisite: MATH& 141 (formerly MATH 121) with a "C" or better or placement test. Back To Top MATH& 146 Introduction to Probability and Statistics5 Credits Introductory probability theory and statistical concepts including organization of data, sampling, descriptive and inferential statistics. Use of probability distributions in parameter estimation, hypothesis testing. Linear regression and correlation. This course was formerly known as MATH 108. Prerequisite: MATH 099 or appropriate testing. Back To Top MATH 147 Precalculus for Business/Social Science5 Credits Properties and applications of elementary algebraic, exponential and logarithmic functions relevant to business, economics and social sciences. Includes matrices, linear inequalities and mathematics of finance. Prepares student for MATH& 148 Business Calculus. This course was formerly known as MATH 156. Prerequisite: MATH 099 with a grade of "C" or better, or placement by testing. Back To Top MATH& 148 Business Calculus 5 Credits An introduction to calculus for students of business and social science. This course was formerly known as MATH 157, Calculus for Business and Social Science. Prerequisite: MATH 147 (formerly MATH 156) or MATH& 141 (formerly MATH 121). Back To Top MATH& 152 Calculus II5 Credits Continuation of MATH&151. The definition, properties, and applications of definite and indefinite integrals. The calculus of inverse trigonometric functions. Techniques of integration. This course was formerly known as MATH 124. Prerequisite: MATH& 151 (formerly MATH 123). Back To Top MATH 205 Linear Algebra5 Credits An introduction to linear algebra for students of science and engineering. Includes vectors in the plane, in three dimensional space, and in-dimensions; matrices and systems of equations, determinants, vector spaces and linear transformations. Prerequisite: MATH& 142 (formerly MATH 122) and MATH& 151 (formerly MATH 123). Back To Top MATH& 254 Calculus IV 5 Credits An introduction to analytic geometry in three dimensions, and vector functions. The calculus of functions of two and three variables and vector functions. This course was formerly known as MATH 126, Multivariable Calculus. Prerequisite: MATH& 151 (formerly MATH 123) and MATH& 152 (formerly MATH 124). Back To Top
My Advice to a New Math 175 Student: With only the final left in 175 I have three types of advice for you. The first one is that you need to keep up on your daily homework. If you miss a days work and fall one day behind, you have to do two days work that next day. Falling behind is detrimental because each day builds off the one before. Staying on top of the homework makes each days new material that much easier to comprehend the first time, and then faster to get the homework done that night. Secondly, try to work with other 175 students. The first time I studied with another student in this class was before the fourth exam and I regret not practicing this way earlier. Studying with another student allows you to ask them questions you are unclear of, and they may have similar questions which both of you can work through. It also allows you to work through problems they may have and you fully understand. Being able to effectively teach a fellow student the material displays how well you know the material yourself. The last piece of advice I have is the hardest thing to get you to do. The exams cover a lot of material and require you to work through them quickly and display that you know the material and need little effort to show it. The exams are challenging in that they require time management. My advice is that you practice and review problems as if you were taking the exam. Work through them quickly showing your work. The books chapter review is an excellent way to do this. Dr. Hoar is an excellent professor and will help you learn the material when you demonstrate that you need help and are wiling to put some effort into your learning of the material. The math tutors are available also, but I feel as though going right to Dr. Hoar is the best way to get your problems solved. If after seeing him questions are still in your mind, then seek the help of tutors or fellow students.
Courses 120. Appreciation of Mathematics An exploration of topics which illustrate the power and beauty of mathematics, with a focus on the role mathematics has played in the development of Western culture. Topics differ by instructor but may include: Fibonacci numbers, mathematical logic, credit card security, or the butterfly effect. This course is designed for students who are not required to take statistics or calculus as part of their studies. 140. Statistics An introduction to statistical thinking and the analysis of data using such methods as graphical descriptions, correlation and regression, estimation, hypothesis testing, and statistical models. A graphing calculator is required. 160. Calculus for the Social Sciences A graphical, numerical and symbolic introduction to the theory and applications of derivatives and integrals of algebraic, exponential, and logarithmic functions, with an emphasis on applications in the social sciences. A student may not receive credit for both Mathematics 160 and 181. 181. Calculus I A graphical, numerical, and symbolic study of the theory and application of the derivative of algebraic, trigonometric, exponential, and logarithmic functions, and an introduction to the theory and applications of the integral. Suitable for students of both the natural and the social sciences. A graphing calculator is required. A student may not receive credit for both Mathematics 160 and 181. 182. Calculus II A graphical, numerical, and symbolic study of the theory, techniques, and applications of integration, and an introduction to infinite series and/or differential equations. A graphing calculator is required. Prerequisite: Mathematics 181 or the equivalent. 201. Modeling and Simulation for the Sciences A course in scientific programming, part of the interdisciplinary field of computational science. Large, open-ended, scientific problems often require the algorithms and techniques of discrete and continuous computational modeling and Monte Carlo simulation. Students learn fundamental concepts and implementation of algorithms in various scientific programming environments. Throughout, applications in the sciences are emphasized. Cross-listed as Computer Science 201. Prerequisite: Mathematics 181. 210. Multivariable Calculus A study of the geometry of three-dimensional space and the calculus of functions of several variables. Prerequisite: Mathematics 182. 212. Vector Calculus A study of vectors and the calculus of vector fields, highlighting applications relevant to engineering such as fluid dynamics and electrostatics. Prerequisite: MATH 182. 220. Linear Algebra The theoretical and numerical aspects of finite dimensional vector spaces, linear transformations, and matrices, with applications to such problems as systems of linear equations, difference and differential equations, and linear regression. A graphing calculator is required. Prerequisite: Mathematics 182. 235. Discrete Mathematical Models An introduction to some of the important models, techniques, and modes of reasoning of non-calculus mathematics. Emphasis on graph theory and combinatorics. Applications to computing, statistics, operations research, and the physical and behavioral sciences. 240. Differential Equations The theory and application of first- and second-order differential equations including both analytical and numerical techniques. Prerequisite: Mathematics 182. 250. Introduction to Technical Writing An introduction to technical writing in mathematics and the sciences with the markup language LaTeX, which is used to typeset mathematical and scientific papers, especially those with significant symbolic content. 260. Introduction to Mathematical Proof An introduction to rigorous mathematical argument with an emphasis on the writing of clear, concise mathematical proofs. Topics will include logic, sets, relations, functions, and mathematical induction. Additional topics may be chosen by the instructor. Prerequisite: Math 182 280. Selected Topics in Mathematics Selected topics in mathematics at the introductory or intermediate level. 310. History of Mathematics A survey of the history and development of mathematics from antiquity to the twentieth century. Prerequisite: Math 260. 410. Geometry A study of the foundations of Euclidean geometry with emphasis on the role of the parallel postulate. An introduction to non-Euclidean (hyperbolic) geometry and its intellectual implications. Prerequisite: Mathematics 260 421 - 422. Probability and Statistics A study of probability models, random variables, estimation, hypothesis testing, and linear models, with applications to problems in the physical and social sciences. Prerequisite: Mathematics 210 and 260. 435. Cryptology An introduction to cryptology and modern applications. Students will study various historical and modern ciphers and implement select schemes using mathematical software. Cross-listed with COSC 435. Prerequisites: MATH 220 and either MATH 235 or 260. 439. Elementary Number Theory A study of the oldest branch of mathematics, this course focuses on mathematical properties of the integers and prime numbers. Topics include divisibility, congruences, diophantine equations, arithmetic functions, primitive roots, and quadratic residues. Prerequisite: MATH 260. 441 - 442. Mathematical Analysis A rigorous study of the fundamental concepts of analysis, including limits, continuity, the derivative, the Riemann integral, and sequences and series. Prerequisites: Mathematics 210 and 260. 445. Advanced Differential Equations This course is a continuation of a first course on differential equations. It will extend previous concepts to higher dimensions and include a geometric perspective. Topics will include linear systems of equations, bifurcations, chaos theory, and partial differential equations. Prerequisite: Math 240. 448. Functions of a Complex Variable An introduction to the analysis of functions of a complex variable. Topics will include differentiation, contour integration, power series, Laurent series, and applications. Prerequisite: MATH 260.
Phoenix Algebra ...Differential equations play a prominent role in engineering, physics, economics, and other disciplines. Differential equations arise in many areas of science and technology, specifically whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and t...
- Lie The Mathematical Sciences in Society~s Service For wise policy decisions, the tasks and needs of the mathematical sciences. discussed in the previous chapter, must be viewed in proper context. A major element of this context is the whole process by which the mathematical sciences contribute to society's ends. To understand this in full detail would be extremely difficult probably impossible but a general overview can be given in reasonable space and with some clarity. THE MATHEMATICAL POPULATION The most significant fact about the people and institutions that employ the mathematical sciences, at one level or another, in our society is their number and diversity. Exact figures on the number of our people who have had at least two years of high school mathe- matics might be hard to find. Rough but probably adequate estima- tion leads to a figure of perhaps one fourth of the nation's adult population. With the increasing complexity of society's mech- anisms, institutions, and interrelations, leading inevitably to greater public education, this fraction is in the process of slowly moving # According to reference 44, 47.7 percent of the U.S. population over 14 years of age has completed four years of high school and 70.5 percent has completed one through three years of high school. Also, 17.6 percent has completed one through three years of college and 8.1 percent of the population over 14 years of age comprises college graduates. 211 OCR for page 212 212 Conclusions upward. It may well reach one third, one half, and even two thirds as the years pass. A fundamental education in mathematics extends through high school and about two years of college. Perhaps 8 to 10 million of our people have this foundation, which required 9 to 11 years of study of mathematics. Not all these people make regular use of what they have learned, but a large fraction of those engaged in physical sci- ence and engineering do. Altogether, the number who use 11 to 13 years of mathematics, at least occasionally, probably runs between 1 and 2 million. Now we look at those who work in mathematical sciences in some way or another. The two largest groups are somewhat over 100,000 high school teachers of mathematics and 200,000 computer pro- grammers, to which we should add the roughly 50,000 members of professional societies dealing with mathematical research, college teaching of mathematics, statistics, computer science, operations re- search, and management science all told, approximately 350,000 to 400,000 people. Another group that needs specific notice includes many workers in agricultural, biological, and medical research, many concerned with production or marketing in industry, most research psycholo- gists, and many workers in other fields of behavioral sciences. Some 50,000 to 100,000 such people use statistical methods in their pro- fessional work. Many look actively to research in statistical method- ology to provide better tools for their use. Finally, toward the tip of the pyramid there are about 7,000 PhD's in the mathematical sciences, among whom over 1,000 are active innovators. This relatively small group bears the responsibility for practically all research in mathematical sciences, for all research education, and for directing much of the college-level education. The fraction of our population contained in any of these groups can confidently be expected to increase. Society's increasing com- plexity and the increasing complexity of its individual mechanisms and institutions will assure this. To estimate the rates of increase, however, is exceedingly difficult, and the figures we now give are only a very rough guess. It seems that the number of people with two years of high school mathematics increases at about 4 percent a year. The pool of those with two or more years of college mathe- matics grows perhaps 8 percent a year-somewhat more rapidly- while pools of those who use college mathematics may grow as much as 12 percent a year. The numbers of those who use mathematical OCR for page 213 The Mathematical Sciences in Society's Service 213 science as a main professional component Stow at diverse annual rates: high school teachers of mathematics at perhaps 5 percent, computer programmers at a rousing 30 percent, professionals at perhaps 14 percent, overall at, say, something over 10 percent. The annual rate of increase of new PhD's over recent years has been 18 percent; as already stated, we believe that active investigators are increasing at about the same percentage Per year. The greater rates of increase at the upper layers of the pyramid are inevitable consequences of the increasing subtlety and complex- ity of society's demands. An 18 percent annual increase at the re- search level is no more than would be expected from the other parts of the picture. We cannot be sure whether the recent rates of growth will or will not continue for the near future. The rate of educating scientifi- cally trained people in each specialty is controlled by students' choices and available facilities. These factors are constantly chang- ing. Over the post-Sputnik decade, enrollments in advanced under- graduate courses in mathematical science grew rapidly, but with a possible tendency to flatten out. This apparent flattening out may be a short-time fluctuation, may represent limited facilities in terms of faculty, may reflect the absence of undergraduate programs in applied fields, or may be due to a partial reorientation of student values from technical and scientific to social concerns. Graduate enrollments in mathematical science, particularly at research-trainir~g levels, are still growing actively, but their future behavior is equally uncertain. If research training in mathematical science continues to expand rapidly, it will be either because of a continuing atmosphere of general public approval or because we shall have opened the way to graduate work to a wider variety of students by removing social obstacles, by establishing a greater di- versity in undergraduate programs, or by broadening understanding and knowledge of mathematical sciences among all college teachers of mathematics. All these reasons for continuing growth are signifi- cant and appropriate. If they produce a continuing expansion, the nation has important tasks for all those who will receive research · . training as a consequence. ~ At the same period the engineering numbers flattened out, physics showed moderately strong increases followed by flattening, geology suffered a severe de- cline followed by a partial recovery, and biology grew, first moderately then more rapidly. OCR for page 214 914 MATHEMATICAL STRATEGY Conclusions To assess the significance of research in mathematical sciences from the national point of view, it is important to remember that, as already described in Chapter 3, the strategy of research in mathe- matics is rather different from that in other sciences. In nuclear or high-energy physics, for instance, a few problem areas are regarded as crucial at any given time. These are confronted in great force by many people. Substantial numbers of groups of reasonable size, each necessarily well supported by machines of various kinds, attack the same problem. This strategy has brought rapid progress to these areas of physics, probably in part because of the relative narrowness of their research objectives. Even physics in general aims at understanding only one universe, under only one system of laws. Mathematical research, seen from society's perspective, has a broader objective: the full development of concepts, results, and methods of symbolic reasoning that will apply to as many as pos- sible of mankind's diverse problems, includin~vitally but far from exclusively the problems arising from the progress of physics. The development of concepts and theories motivated by the needs of mathematics itself must be part of this objective since experience shows that these may well become crucially important for applica- tions. As a result, mathematics must contribute to understanding diverse situations under widely different systems of laws. Mathematical sciences must eventually travel many roads. All past experience, from the dawn of history to recent times, teaches us that the ultimate applicability of a mathematical concept or technique can hardly ever be predicted, that only quite short-range forecasts can be trusted, and that calls for massive effort at one point, at the expense of efforts at other points, should usually be resisted. In this situation, mathematical sciences proceed by a large num- ber of small independent research efforts, often conducted by single individuals or by small groups of men. A large variety of problems is attacked simultaneously. The choice of problems to work on, as in all sciences, is one of the things that determines the success or failure of an investigator. But the mathematician and the mathe- matical scientist have, and need, great freedom in making these choices. This strategy has proved successful. It involves dispersal of forces and active work in many seemingly disconnected fields. Thus there OCR for page 215 The Mathematical Sciences in Society's Service 215 have been repeated periods of apparent overspecialization when mathematicians seemed to be drawing too far from one another (most recently in the 1930's and 1940's). Every time this has hap- pened, however, an apparently inherent unity of mathematics has shown itself again through the appearance of new and more gen- eral concepts and approaches that, as in the present decade, have restored to mathematics much more in the way of unity than had seemed possible a few decades earlier. TRANSFER TIMES When the British steel industry was Denationalized in 1967, the new chairman warned the British people not to expect too much too soon, saying that, in a "capital-intensive" industry, only slow change could be expected. Our nation's system of mathematical service is an institution in which change must be even slower. Indeed, this is a "training-intensive" institution; its greatest investment is in people with years of training. The building of steel plants can be greatly accelerated, but no large group of people can be given 10, or 15, or even 20 years of continuous training in appreciably less than that number of years. As a language, used for communication both with others and with oneself, mathematics shares with the language of words the need for a long and arduous apprenticeship. Including mathemati- cal training in elementary and secondary school, a college graduate majoring in mathematics has typically studied mathematics about three times as long as a college graduate majoring in a science has studied his or her science. The proper time scale for thinking about our society's system of mathematical science is not merely long, as it must be for all the sciences, but very long. From society's viewpoint, the largest reason for supporting self- motivated research in mathematical sciences is the continuing im- pact of the resulting innovations, first as immediate mathematical applications and then more broadly. How fast ought society to ex- pect the results of innovation to be transferred? Surely not in a day or a week or a month. But in one year, or three, or ten? We have stressed the differences between the strategy of most mathematical research and that of the other sciences and tech- nologies. The individual character of the work and the difficulties of forecasting where progress will prove most important have led to OCR for page 216 216 Conclusions a spreading out of attention over a wide variety of problems. (Im- portant mathematical problems, like many of those posed by David Hilbert in a celebrated 1900 address, are often under attack for several decades before their final solution.) As a consequence of its implicit strategy of pressing ahead wherever it seems that reasonably valuable ground can be gained, mathematical science sometimes prepares the way very far in advance. That important uses should follow discovery by decades, often by several decades, should neither be a cause for surprise nor a reason for criticism. It is a state of affairs intrinsic in an efficient use of human resources and the facilities and money that support them. A bare trace of our progress in inner-directed research will come to use in one year. A little more will come in three years. A sub- stantial fraction of what will contribute outside mathematical sci- ences will have begun to make its contribution in 10 years, but only a substantial fraction and only as a beginning. We will do well to do whatever we reasonably can to speed up the process of transfer to use, for there are real gains to be had if we can, but we dare not delude ourselves that great gains in speed can be had by some drastic rearrangement of activity and interest. The mathematical strategy of widespread attack in small parties is weI1 adapted to both the subject and the demands for innovation laid upon it by the diverse needs of society: long delays in transfer to use are an inevitable consequence of this strategy. In planning support of mathematical sciences, especially support of inner- directed research, we must take the long view if our programs are to contribute to the demands that society will make at times spread through the future. We do many other things today with a hope of social gain over decades. The elementary and high school education of youth is to be of value to them and to society, not just for a decade or two, but for four or five or even six decades. All our affairs cannot be con- ducted in the way a field of corn is tended, plowed under this year but reseeded for next year. Innovation in science is more like an apple tree; 10 to 25 years are needed for the crop to return, many- fold, the effort of planting, grafting, and cultivation. Mathematical sciences call for time scales even longer than do other sciences. OCR for page 217 The Mathematical Sciences in Society's Service 217 EMERGENCIES There is an exception to the usual need for long transfer times typical in the mathematical sciences the use of creative mathe- maticians during emergencies. World War II generated many technical emergencies. Mathe- maticians usually involved in inner-directed research responded to many calls: how to conduct antisubmarine warfare, what principles to use in fighter and bomber gunsights, and many questions in ballistics, radar, atomic weapons, and cryptography. The crisis was clear; insight, knowledge, and skill were freely mobilized. Many concrete problems whose solution was explicitly demanded were attacked powerfully and effectively by mathematicians, precisely because of their professional ability and training in thinking into the heart of a problem and seizing on its essentials. And the free- wheeling instinct of the self-motivated researcher played its part. Another emergency arose slowly and imperceptibly in mathemati- cal education, primarily in elementary schols and high schools. The urgent need for reform became apparent about 15 years ago. Leader- ship in meeting this emergency came in a large measure from uni- versity and college mathematicians active in inner-directed research. (There are various opinions in the mathematical community about the success of the reform movement thus far; there is no dispute about the necessity of curriculum reform.) When the next major change in mathematical education comes, leadership will again have to be drawn from those concerned with inner-directed research. Mathematical scientists who give most of their time to inner- directed research are an important national resource in emergencies. This sort of resource serves special purposes in an emergency, but it cannot be used for these purposes steadily. This type of value has to be a by-product. As long as emergencies continue to arise at the usual rate, society can count on this resource created by basic mathe- matical research as one sort of return from its investments in re- search training in mathematical science. It can do this, however, only by using research-trained people quite differently most of the time. THE LEVEL OF INVESTMENT We are now in a position to ask whether the total investment in basic mathematical research has been at a reasonable level. Let us OCR for page 218 218 Conclusions follow our discussion of Level and Sources of Support (see page 163) and assume federal support of basic research in the universities as $35 million in 1966. At an 18 percent rate of growth, the total through all the past would be about six times as large as the present annual amount, amounting to $200 million. Allowance for non- federal support and for slower rates of growth in the past might raise this to $250 million. If we believe in compound interest, and insist on totaling up present values of all sums spent in the past, this figure would be roughly $300 million. Thus, if we want a total investment picture, we can say that the whole U.S. investment in research in the mathematical sciences to date is about as much as we will spend as initial capital investment on the new super-large "atom-smasher" now approved. When we look at this investment in mathematical research as contributing in diverse and important ways to the effectiveness of the whole na- tional system of mathematical service, where some of these ways have begun, others are beginning, and others will start at times spread forward through decades, and where most contributions will continue for a long time, our investment seems conservative and cautious, perhaps disproportionately small. GROWTH CANNOT BE FOREVER An 18 percent a year increase means doubling every four years. A 10 percent annual increase means doubling in less than 10 years. Such doubling cannot continue indefinitely. Not only mathematical science but all science and all technologies with growing research sectors must face the need for an ultimate tapering off. Neither the fraction of gross national product that can be devoted to research nor the number of people potentially capable of becoming research investigators can increase indefinitely. At the moment, the need for the innovations of mathematical- science research is large and growing. These needs would requir continuing rapid expansion for the near future, even without the necessity of continuing the mathematical teaching of nonprofes- sionals. But what of the day when tapering off of growth becomes appropriate? What will be the environment, the pressures, the appropriate adjustments? Education above the high school level is in transition. The clas- sical division between undergraduate and graduate work is more and more clearly seen to be at an inappropriate place. We may be OCR for page 219 The Mathematical Sciences in Society's Service 219 moving toward a three-part scheme of scientific education, consist- ing of (1) the present freshman and sophomore years, (~) the last two undergraduate years and graduate study through the compre- hensive examination, and (3) thesis research and postdoctoral train- ing. It will be after this pattern has appeared more clearly that we shall have to face declining rates of increase in supported research. At present, five sixths of new PhD's follow career patterns that do not lead to federally supported research. If our needs for profes- sionally trained personnel were to cease to rise, adjustments would be fairly simple. If, as seems more likely, the future demand for professionally trained personnel increases much more than in pros portion to the amount of sponsored research that the nation can afford, there will have to be major readjustments. Direct contact with research leadership contributes greatly to education in mathematical sciences below the research level. In the future, research leaders may not be able to contribute much time to this activity. Once this situation arises, forms of mass communi- cation, both television and film, will become important in the process, and there will have to be real innovations designed to pro- vide face-to-face contact with many students. Inevitably, the selection of a principal investigator for research support will focus increasingly on the number and quality of his PhD candidates, as well as on the number and quality of his own contributions. As a consequence, it will become clearer that academic tenure exists primarily for teaching rather than for research sup- port. Such times, when they come to each science, will be times of change indeed. There are many reasons, however, why they will not come to all sciences together. The bigger the initial budget, the sooner its doubling will bring it to a point where things must be done. The larger the system of activities that research in a science supports, the larger the total the nation can wisely spend on it. Today, academic mathematical research receives a relatively small amount of support less than three cents out of every dollar spent by the federal government for research at universities. Yet this mathematical research is the leading wedge of a very large national effort. Thus, reduction of growth rate in mathematical research ought, wisely and advisedly, to take place only after similar reduc- tions have been made in many other fields. Thus the mathematical sciences can expect much guidance by the time they are faced with making their own adjustments.
Location This is the second mid-term of the course. It involved the use of more advanced Matlab programming. This includes LU decomposition, plotting, fitting data, Gauss-quadrature, solution to non-linear equations, and making functions.
The EPGY Calculus B course covers much of what is generally covered in a second-quarter college calculus course. Together with Calculus A, this course prepares students to take the AB Calculus AP exam. The course is designed to take about 12 weeks to complete on average. Prerequisite: M040 or equivalent.
The Use Of Technology Mathematics is one of those rare disciplines which has an air of timelessness about it, and because its truths are eternal many people think that both mathematics and the way we teach it has remained unchanged over the years. Little could be further from the truth, however - the nature of how we teach and investigate mathematics has changed a lot over the years, particularly since the invention of the calculator and computer. The Department of Mathematics at Southeastern makes use of technology in many different ways - from graphing on calculators in the classroom to using the Web in our courses to the production of the CD you're using right now. Much of what we use falls into 3 main categories: graphing calculators, the Worldwide Web, and the computer algebra system Mathematica. To see these radically different types of technology, follow the following links: One thing to keep in mind is that although we use technology, we use it carefully and only where we think it will enhance our students' understanding. At many schools there is a drive to use technology because it is the "latest thing to do", and it can end up eclipsing the mathematics rather than enhancing it. At Southeastern we keep in mind that technology is a great tool in learning, but it's just that - a tool. An important tool, but just one part of the process of learning, understanding, and doing mathematics.
The first analysis course I ever took used the book "Elementary Analysis" by Ross. It's basically baby-baby-Rudin. Ross's book (if you include the exercises and the optional sections) covers more or less the same material as the first 8 chapters of baby-Rudin, but the exposition is much friendlier and it's more easy-going for a beginner. When I say "beginner" here I really mean beginner -- someone who has never even written a rigorous mathematical proof. The book would probably be very boring and tedious for someone above this level.
The Algebraic Concepts Unit includes Competencies/Objectives which focus on algebraic equations and operations. This unit includes studying number systems, operations, and forms. Students explore the symbolic nature of algebraic concepts by identifying and extending patterns in algebra, by following algebraic procedures, and by proving theorems with properties. The Geometry Unit includes Competencies/Objectives which focus on exploring geometric concepts from multiple perspectives. The Geometry Unit includes properties and construction of figures, proofs and theorems, history of geometry, transformations, logic, and problem solving.
Place Value and Names for Numbers. Addition with Whole Numbers, and Perimeter. Rounding numbers, Estimating Answers, and Displaying Information. Subtraction with Whole Numbers. Multiplication with Whole Numbers, and Area. Division with Whole Numbers. Exponents and Order of Operations. Summary. Review. Test. Projects. 2. FRACTIONS AND MIXED NUMBERS. Meaning and Properties of Fractions. Prime Numbers, Factors, and Reducing to Lowest Terms. Multiplication with Fractions, and the Area of a Triangle. Division with Fractions. Addition and Subtraction with Fractions. Mixed-Number Notation. Multiplication and Division with Mixed Numbers. Addition and Subtraction with Mixed Numbers. Combinations of Operations and Complex Fractions. Summary. Review. Cumulative Review. Test. Projects. softcover teacher edition with all Students content and solutions text only no supplement ship immediately - Expedited shipping not available oversized $6.59 +$3.99 s/h Good BookSleuth Danville, CA Fast Shipping ! Used books may not include access codes, CDs or other supplements. $69.95 +$3.99 s/h Good a2zbooks Burgin, KY Has some markings. Has minor shelf and corner wear. Has minor corner curl and creasing to covers. Has water stain at back of book, does not affect text. Binding is in good condition. Has stickers on s...show morepine and back cover. Quantity Available: 1. Category: Mathematics; Math/Algebra/Trig; ISBN: 0495013919. ISBN/EAN: 9780495013914. Inventory No: 1560788120. 2nd Edition. ...show less $69.95 +$3.99 s/h Good A2ZBooks Ky Burgin, KY Pacific Grove, CA 2006 Softcover 2nd Edition Good Condition Has some markings. Has minor shelf and corner wear. Has minor corner curl and creasing to covers. Has water stain at back of book, does no...show moret affect text. Binding is in good condition. Has stickers on spine and back cover. Quantity Available: 1. Category: Mathematics; Math/Algebra/Trig; ISBN: 0495013919. ISBN/EAN: 9780495013914. Inventory No: 1560788120
A set for students that are ready for advanced algebra skills. This series is geared for high school level students. Topics include: Solve for the Unknown (Using logs), Solving Inequalities by Adding and Subtracting, Solving Inequalities by Multiplying and Dividing, Solving Fractional Equations, Absolute Value Equations, Slope of a Line, Slope and Equation of Lines, Algebraic solution to linear system, Algebraic Solutions to Simultaneous Equations, Algebraic Translations
Trigonometry new 2nd edition of Cynthia Young's Trigonometry continues to bridge the gap between in-class work and homework by helping students overcome common learning barriers and build confidence in their ability to do mathematics. The text features truly unique, strong pedagogy and is written in a clear, single voice that speaks directly to students and mirrors how instructors communicate in lectures. The new second edition of Cynthia Young's Algebra & Trigonometry continues to bridge the gap between in-class work and homework by helping students overcome common learning barriers and build confidence in their ability to do mathematics. Now in its 10th edition, Analytic Trigonometry is a book that students can actually read, understand, and enjoy. To gain student interest quickly, the text moves directly into trigonometric concepts and applications and reviews essential material from prerequisite courses only as needed.
Haymarket Algebra getting to know your specific need and to help you achieve your objectives. Respectfully Yours, John B., BSME, FE(EIT), MBA, PMP, ITILv3, CSSGBEngineers & Scientists use Mathematics to communicate as much as divers use air to breath. As an Engineer myself, I've come to realiz...Most of my family is deaf, including both parents and both step parents. I also have a sister who is deaf along with her husband and her two kids. My other sister who is hearing, is a sign language interpreter. ...Hence, much will depend on student?s standing. The following is a snapshot of what will be covered in the course: algebraic expressions, setting up equations by translating word problems; evaluating expressions by adding and subtracting polynomials; factoring polynomials (trinomials) using FOIL ...
For many years, this classroom-tested, best-selling text has guided mathematics students to more advanced studies in topology, abstract algebra, and real analysis. Elements of Advanced Mathematics, Third Edition retains the content and character of previous editions while making the material more … Starting with the most basic notions, Universal Algebra: Fundamentals and Selected Topics introduces all the key elements needed to read and understand current research in this field. Based on the author's two-semester course, the text prepares students for research work by providing a solid … Accessible to all students with a sound background in high school mathematics, A Concise Introduction to Pure Mathematics, Third Edition presents some of the most fundamental and beautiful ideas in pure mathematics. It covers not only standard material but also many interesting topics not usually … Retaining all the key features of the previous editions, Introduction to Mathematical Logic, Fifth Edition explores the principal topics of mathematical logic. It covers propositional logic, first-order logic, first-order number theory, axiomatic set theory, and the theory of computability. The … Shows How to Read & Write Mathematical ProofsIdeal Foundation for More Advanced Mathematics Courses Introduction to Mathematical Proofs: A Transition facilitates a smooth transition from courses designed to develop computational skills and problem solving abilities to courses that emphasize … Introduction to Fuzzy Systems provides students with a self-contained introduction that requires no preliminary knowledge of fuzzy mathematics and fuzzy control systems theory. Simplified and readily accessible, it encourages both classroom and self-directed learners to build a solid foundation in … A First Course in Fuzzy Logic, Third Edition continues to provide the ideal introduction to the theory and applications of fuzzy logic. This best-selling text provides a firm mathematical basis for the calculus of fuzzy concepts necessary for designing intelligent systems and a solid background for … This introductory graduate text covers modern mathematical logic from propositional, first-order and infinitary logic and Gödel's Incompleteness Theorems to extensive introductions to set theory, model theory and recursion (computability) theory. Based on the author's more than 35 years of teaching … … Computability theory originated with the seminal work of Gödel, Church, Turing, Kleene and Post in the 1930s. This theory includes a wide spectrum of topics, such as the theory of reducibilities and their degree structures, computably enumerable sets and their automorphisms, and subrecursive …
Sixth Form Maths AS and A Level Mathematics - Edexcel This course is suitable for anyone having studied a Higher level GCSE, although students who achieve less than a grade B may find the course very difficult. The course provides an excellent balance of mathematical topics included within the areas of Pure Mathematics (Core), Mechanics and Statistics. The subject is allocated five hours tuition time a week in years 12 and 13. Mathematics is a very popular subject in the sixth form and results are excellent. A Level Further Mathematics This course is studied as an additional A level in Mathematics for students who have a real love for the subject, and who have achieved an A* grade at GCSE. It is increasingly being seen as a useful addition for students who are keen to study Mathematics or Engineering at University, as well as other Mathematics related subjects. The course is similar to the A level in Mathematics, with exams in the areas of Pure Mathematics (Core), Mechanics, Decision Maths and Statistics. The subject is allocated four hours tuition time a week in Years 12 and 13. AS level Further Maths For students who would like to settle in before taking Further Maths, there is the option of an AS level in the subject in Year 13 if the student achieves a grade A in their AS level Mathematics. This will be studied alongside Maths and will require students to sit three extra modules. GCSE Re-takes Students wishing to improve their GCSE grade in Mathematics will be entered for the OCR graduated assessment GCSE. This comprises two modular exams, and a terminal paper. Facilities The department is well funded and has a good stock of textbooks. The main textbooks used are written by the syllabus examiners and moderators themselves. We also have a range of software that can be accessed by the students at any time. Graphic programmable calculators are highly recommended. The department prefers students to have their own and sells them at a very reasonable price during the first term. Staff Sixth form teachers enjoy their Mathematics and the challenge of helping students achieve their maximum potential. Some reasons to study Mathematics (and Further Mathematics) at A Level It is a highly desirable subject If you want to go to University then A level Mathematics will open more doors than any other subject: Courses and careers in Mathematics, Engineering, Physics, Computing, Accountancy, Economics, Business, Banking, Air Traffic Control, Retail Management, Architecture, Surveying, Cartography, Psychology and, of course, Teaching to name but a few. In fact, for entry to a Business Studies degree, institutions will generally look for Mathematics as their first choice, and it may seem strange, but if you want to study any area of computing, including games design, at University then they generally ask for Mathematics A Level ahead of Computing A Level. Even subjects such as Psychology favour a qualification in Mathematics. Mathematics is everywhere The world we live in simply would not exist as we know it if it was not for Mathematics. Mobile phones, computers, all wireless technologies, computer games, special effects in films, bypasses, CAT scanners, weather forecasts… all of these things have mathematics at the heart of their technology, and the companies who develop these employ many mathematicians. If you were to Google all the creators of The Simpsons, you will find they all have Mathematics based degrees and PhDs. Incidentally, Google itself was created by Mathematicians. You will earn more if you study Mathematics A level Students who study Mathematics A Level on average earn 10% more than their non-Mathematics studying counterparts. This percentage increases if you go on to study Mathematics at degree level.
MATERIALS NEEDED FOR CLASS 1. Notebook. (3 ring binder with paper) 2. Pencil for quizzes/tests. (pen may be used for notes/worksheets) 3. Textbook with book cover. 4. Graphing calculator. (TI-83+ or TI-84+ recommended) GRADING SCALE 95-100 A 93-94 A- 91-92 B+ 87-90 B 85-86 B- 83-84 C+ 79-82 C 77-78 C- 75-76 D+ 72-74 D 70-71 D- 0-69 F ATTENDANCE Absences: If you are absent, please get assignments by one of the following methods: 1. asking me when you return to school. 2. calling a reliable friend from class. 3. checking the Milwaukee Lutheran website If you are absent for a long period of time, please contact the school so that assignments can be sent home. You are required to make up the missing work. When you get back, you need to make arrangements with me to get the work completed. If an absence is due to a field trip, vacation or other school activities, you must get assignments ahead of time and make arrangements for missing quizzes or tests ahead of time. Tardies: You must be completely in the classroom by the end of the tone. Please get seated immediately. HOW TO BE SUCCESSFUL 1. Take notes (in a notebook) during class. Copy sample problems from the board and try them as we go through them in class. Do the assignments in your notebook. If you have trouble with the assignment problems, come in and get extra help. All worksheets must be completed! Everyone must complete a notebook for the first two chapters and submit these at the end of the chapter. 2. Use all given class time to begin your practice problems. 3. Take announced quizzes. 4. Take Chapter Tests. a. Chapter tests are cumulative. The last cumulative chapter test of the semester will be the final exam. 5. Complete all projects. a. Projects will be given to show application to real life situations. These projects are intended to extend your learning and to allow you to utilize skills beyond mathematics. b. Projects may be turned in early for a preliminary check(s) and then corrected. c. Additional information will be given prior to beginning a project. NOTE: All students will be required to complete and submit a notebook for the first two chapters. After that, any student whose average falls below a C- (77%) will be required to complete and submit all homework/notes in an organized notebook. ON-LINE GRADES To access grades go to the Milwaukee Lutheran website. Daily work will be posted within 2-3 days. Quizzes/Tests will be posted within 1 week. MISCELLANEOUS Cheating Policy: You will receive a zero along with the person who gave you the answers or work and will be referred through the disciplinary cycle. Extra Help:Hours 3 and 8 (with a pass from me) before school (7:35-7:50 Mondays; 7:20-7:50 Tuesdays, Wednesdays, Fridays) before school (7:30-8:30 on Student Help Thursdays) after school (3:06-3:35 ) (except when coaching) Passes:Given for emergencies only! (must have planner and school ID) Classroom R's:Responsibility and Respect You are responsible for your education. Others, like myself, are here to guide and help you. Please respect the right of others wanting to learn.
Pre-Algebra is the link between basic math and Algebra. We will be studying positive and negative integers, and learning how to solve equations for unknown variables. Get ready and buckle up... it's going to be one heck of a ride!
Math Curriculum This course is taken predominately by seventh graders. It can also be taken by new eighth graders to Woodward Academy who have not successfully completed a pre-algebra course. The course includes the following content: Integers and Algebraic Expressions Equations and Inequalities Graphing in the Coordinate Plane Real Numbers Applications of Proportions Applications of Percent Exponents and Powers Geometry Geometry and Measurement Using Graphs to Analyze Data Probability Algebraic Relationships enhanced college preparatory course. The students are not expected to do as much work independently. This course is for those students who might be expected to have difficulty with a standard pre-algebraPre-Algebra Honors Text: McDougal Littell Pre-Algebra Students are selected for honors math on the basis of proven exceptional ability, grades in previous mathematics courses, and demonstrated ability to carry the added responsibility of such a course. The honors text is covered at a more rapid pace and to greater extent than the preparatory text. Honors students are required to do exploration problems which extend the concepts presented in each lesson. The teacher also provides supplemental activities and problems from various sources for the student to explore. The course includes the following content: Variables, Expressions, and Integers Solving Equations Multi-Step Equations and Inequalities Factors, Fractions, and Exponents Rational Numbers and Equations Ratio, Proportion, and Probability Percents Linear Functions Real Numbers and Right Triangles Measurement, Area, and Volume Data Analysis and Probability Polynomials and Nonlinear Functions Angle Relationships and Transformations Algebra I Enhanced College Preparatory, College Preparatory, Transition Text: Prentice Hall Algebra I This course will be taken by eighth graders. It is basic to the understanding of further mathematics. The course includes significant work in the areas of statistics, probability, and geometry. It also includes work with scientific calculators and computers. The course includes the following content: Tools of Algebra Solving Equations Solving Inequalities Solving and Applying Proportions Graphs and Functions Linear Equations and Their Graphs Systems of Equations and Inequalities Exponents and Exponential Functions Polynomials and Factoring Quadratic Equations and Functions Radical Expressions and Equations Rational Expressions and Functions Algebra I – Enhanced College Preparatory. The students are not expected to do as much work independently. This course is for those students who might be expected to have difficulty with a standard first-year algebraAlgebra I Honors The students are selected on the basis of proven exceptional ability, grades in previous math courses, and the ability to carry added responsibility of such a course. Teaching instruction and expected work are carried at a more rapid pace and to a greater depth than in the enhanced college preparatory algebra class. The Honors students are required to do exploration problems which extend the lesson content, including experiments, research, and projects. Supplemental enrichment activities are used through the year when appropriate. The course includes the following content: Fractions and Fractals Data Exploration Proporational Reasoning and Variation Linear Equations Fitting a Line to Data Systems of Equations and Inequalities Exponents and Exponential Models Functions Transformations Quadratic Models Probability Introduction to Geometry Geometry Honors Students are selected on the basis of proven exceptional ability, mastery of objectives presented in Honors – Algebra I, and the ability to carry added responsibility of such a course. Teaching instruction and expected work are identical to that presented in the upper school geometry course. The course includes the following content: Points, Lines, Planes, and Angles Deductive Reasoning Parallel Lines and Planes Congruent Triangles Quadrilaterals Inequalities in Geometry Similar Polygons Right Triangles Circles Constructions and Loci Areas of Plane Figures Areas and Volumes of Solids Coordinate Geometry Transformations Calculator Usage (all courses) The TI-30X IIS calculator is the only approved calculator for use on middle school homework and graded assignments. The teacher will define for you, specifically for his or her class, what constitutes appropriate and inappropriate use of the calculator on graded assignments. The teacher reserves the right to: • Have graded assignments on which NO CALCULATOR usage is allowed – even for students with accommodation for calculator use. (For example, if computation is the only skill being tested, then use of a calculator would be an inappropriate accommodation.) • Require students to clear/reset the memory of the calculator before graded assignments. It should be understood without saying, but any use of the calculator's storage abilities to merely create notes/reference materials for use on a graded assignment is obviously a violation of the honor code. The use or even mere presence of such material on a student's calculator during a graded assignment may subject that student to honor council proceedings. The student is responsible for the material stored in the calculator IN HIS/HER POSSESSION, whether or not it is their own calculator.
Basic Math and Pre-Algebra Workbook For Dummies Begin to: Complete with lists of ten alternative numeral and number systems, ten curious types of numbers, and ten geometric solids to cut and fold, Basic Math and Pre-Algebra WorkbookFor Dummies will demystify math and help you start solving problems in no time! Customer Reviews: Exactly what you need By SMF Joubert "Juniper" - July 8, 2009 I was cautious selecting this title- the other reviews weren't too helpful, but I am so glad I chose this specific publication. I barely passed high school math and have all the usual math baggage associated with years of miserable stress and failure (add to that a father who was a professor of applied mathematics and mix in some exceptionally clever siblings). Now as a 34 year old, I wanted to see if I could use the logic skills I acquired as an adult and dabble in some vacation time algebra. I didn't want to study math in the sense of grasping long explanations, I wanted to see if I could do the math. I knew I would need explanations if I wanted to work out the sums, so I wasn't sure which book to choose- the study book or the workbook. The workbook provided me with exactly what I needed. Brief, but thorough explanations with examples and several questions of each type to work out. Even the answer section has step-by-step explanations and it was impossible not to get your head... read more Great biik By Leonardn O. Norris - November 23, 2008 This book has a lot of good information for the beginer who wants to understand Pre algebra it tells you step by step and is easy to read and understand I highly recommend this book to any one who needs help in algebra or is needing to review and learn their algebra skills again. refresher By returning student "returning student" - February 5, 2009 If you only need a refresher and simple, understandable explanations- this book is perfect. I haven't formally done any of this for over 20 years- but this was very helpful. Get the confidence and the math skills. you need to get started with calculus!. Are you preparing for calculus? This easy-to-follow, hands-on workbook helps you master basic pre-calculus concepts and ...
Algebra (from Arabic al-jebr meaning "reunion of broken parts") is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising ... Algebra (from Arabic al-jebr meaning "reunion of broken parts") is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising ... Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part ... Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry arose independently in a number of early cultures as a ... Geometry is the branch of the mathematics that deals with the features and relations of points, lines surfaces, solids and other high dimensional objects. Generally we deal with two types of geometry ...
New with no dust jacket Key Curriculum Press Paperback 4to 11" - 13" tall; 162 pages; A Key Curriculum book, quality paper, punched for three-hole notebook. This book of engaging blackline masters provides activities for algebra students to use with the graphing calculators and graphing software-technology which is rapidly becoming commonplace in the high school math classroom. Creating graphs is no longer a time consuming task for students, which leaves them more time to use graphs to study the properties of functions. Graphic Algebra helps develop new insights into algebra by providing easy-to-use lessons in which students graph and study functions using any graphing calculator or computer software for graphing. The book helps students use graphs to solve problems set in real-world contexts; to link different representations in order to move easily between tables of values, algebraic expressions, and graphs; to develop understanding of different types of functions and their properties; to learn concepts and skills needed for graphing on a calculator or a computer; and to explore transformations of functions. This book grew out of a research project conducted at the University of Melbourne, Australia. Graphic Algebra was designed to be used in a variety of ways to supplement and complement the teaching of algebra. Some problems can be used to introduce new ideas; others offer a novel way to review familiar ideas in a new context. The book is a perfect supplement for any curriculum involving algebra. The materials assume that students have a basic familiarity with algebraic notation and the Cartesian plane. Other prerequisite knowledge is noted for each chapter. Teachers can select short or long sequences of work designed for students at various levels. The book contains reproducible blackline masters, as well as teaching suggestions for using graphing calculators in algebra, extensive teacher notes, and appendices with specific instructions for the Texas Instruments TI-82 and TI-83, Hewlett-PackardŽ HP-38G, and CasioŽ CFX-9850G graphing calculators. For grades 8-11.; ISBN: 1559532793 Seller's Terms of Sale: If you want to use a coupon or promotion you found on our site at Merchant Circle, Twitter, a blog, barbsbooks.com or a magazine ad, put the promotion or coupon code in the section for "Comments" on the order form. The shopping cart total will not reflect the promotion, but we will apply the savings or free item when we process your order. We sell through two different store names on this site with two separate shopping carts -- Barb's People Builders and Barb's People Builders Teaching Help. 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The Algebra 2 Tutor DVD Series teaches students the core topics of Algebra 2 and bridges the gap between Algebra 1 and Trigonometry, providing students with essential skills for understanding advanced mathematics. This lesson teaches students how to simplify expressions that contain radicals. Students are taught how to decompose the radical and pick groupings that can then be pulled out which simplifies the expression. Grades 8-12. 28 minutes on DVD.
... read more Mathematical Modelling Techniques by Rutherford Aris "Engaging." — Applied Mathematical Modelling. A theoretical chemist and engineer discusses the types of models — finite, statistical, stochastic, and more — as well as how to formulate and manipulate them for best results. Introduction to Vector and Tensor Analysis by Robert C. Wrede Examines general Cartesian coordinates, the cross product, Einstein's special theory of relativity, bases in general coordinate systems, maxima and minima of functions of two variables, line integrals, integral theorems, and more. 1963 edition. Vector Analysis by Homer E. Newell, Jr. This text combines the logical approach of a mathematical subject with the intuitive approach of engineering and physical topics. Applications include kinematics, mechanics, and electromagnetic theory. Includes exercises and answers. 1955 edition. Dimensional Analysis: Examples of the Use of Symmetry by Hans G. Hornung Derived from a course in fluid mechanics, this text for advanced undergraduates and graduate students employs symmetry arguments to illustrate the principles of dimensional analysis. 2006 edition. Vectors and Their Applications by Anthony J. Pettofrezzo Geared toward undergraduate students, this text illustrates the use of vectors as a mathematical tool in plane synthetic geometry, plane and spherical trigonometry, and analytic geometry of 2- and 3-dimensional space. Flow-Induced Vibrations: An Engineering Guide by Eduard Naudascher, Donald Rockwell Graduate-level text synthesizes research and experience from disparate fields to form guidelines for dealing with vibration phenomena, particularly in terms of assessing sources of excitation in a flow system. 1994 edition.About Vectors by Banesh Hoffmann No calculus needed, but this is not an elementary book. Introduces vectors, algebraic notation and basic ideas, vector algebra, and scalars. Includes 386 exercises. Product Description: Numerous exercises appear throughout the text. 1962
further. In particular, the place-value numeration system used for arithmetic implicitly incorporates some of the basic concepts of algebra, and the algorithms of arithmetic rely heavily on the "laws of algebra." Nevertheless, for many students, learning algebra is an entirely different experience from learning arithmetic, and they find the transition difficult. The difficulties associated with the transition from the activities typically associated with school arithmetic to those typically associated with school algebra have been extensively studied.1 In this chapter, we review in some detail the research that examines these difficulties and describe new lines of research and development on ways that concepts and symbol use in elementary school mathematics can be made to support the development of algebraic reasoning. These recent efforts have been prompted in part by the difficulties exposed by prior research and in part by widespread dissatisfaction with student learning of mathematics in secondary school and beyond. The efforts attempt to avoid the difficulties many students now experience and to lay the foundation for a deeper set of mathematical experiences in secondary school. Before reviewing the research, we first describe and illustrate the main activities of school algebra. Previous chapters have shown how the five strands of conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition are interwoven in achieving mathematical proficiency with number and its operations. These components of proficiency are equally important and similarly entwined in successful approaches to school algebra. The Main Activities of Algebra What is school algebra? Various authors have given different definitions, including, with "tongue in cheek, the study of the 24th letter of the alphabet [x]."2 To understand more fully the connections between elementary school mathematics and algebra, it is useful to distinguish two aspects of algebra that underlie all others: (a) algebra as a systematic way of expressing generality and abstraction, including algebra as generalized arithmetic; and (b) algebra as syntactically guided transformations of symbols.3 These two main aspects of algebra have led to various activities in school algebra, including representational activities, transformational (rule-based) activities, and generalizing and justifying activities.4 The representational activities of algebra involve translating verbal information into symbolic expressions and equations that often, but not always, involve functions. Typical examples include generating (a) equations that represent quantitative problem situations in which one or more of the quan-
Mastering the methods is more important in the long run than simply being able to do the problems sometimes. It's great if you can solve a problem in multiple ways, but most of them don't work in all cases. They might work for that one specific problem, but they might also ONLY work for that one specific problem. It's best to master the method that works in all, or the most, cases. When it's mastered, you're guaranteed to be able to solve a problem of that given form. If you practice instead a bunch of solutions that only work in a few different cases and don't learn to apply the general solution in many ways, you might eventually end up with a problem you can't solve. tl;dr The goal is to be able to solve any problem of a given type using a general solution that works with all problems of that given type, not to be able to solve specific problems of a given type with solutions that work only for specific problems of that given type.
Middle school > Course Description math Mathematics in grades 6-8 is a sequential, college preparatory program. It emphasizes the development of math concepts, computational skills, problem solving, and critical thinking. Comprehensive and appropriately challenging, this curriculum is designed to provide students with the math background necessary for subsequent math coursework. Math 6 (Grade 6) is a continuation of the Progress in Mathematics program used in the Lower School. The continuity of the program helps to ease the Middle School transition and allows the students to expand their mathematical ability. Concepts including numeration, operations, computation, algebra, functions, geometry, measurement, and probability are still presented in a variety of formats to develop higher level critical thinking. Many skills directly foreshadow pre-algebra. Math 7 (Grade 7) integrates applied arithmetic, algebra, and geometry, and connects all these areas to measurement, probability, and statistics. This course provides opportunities for students to visualize and demonstrate concepts with a focus on real-world applications. A strong algebraic influence is included to prepare students for Pre-Algebra. The text for this course is Mathematics: Course 2 by Prentice-Hall. Pre-Algebra (Grades 7-8) reviews the basic computation of real numbers while integrating skills requiring higher levels of thinking. The use of variables throughout prepares for expanded operations required in Algebra I. Algebra-thinking activity labs provide students with opportunities to dig deeper and explore algebraic concepts to build conceptual understanding. This course, normally taught to eighth graders, is also offered to seventh graders who have demonstrated above average quantitative aptitude and skill. The text for this course is Pre-Algebra by Prentice-Hall. Algebra ICP (Grades 8-9) is offered to all students who have completed Pre-Algebra. It extends the concept of set theory to include algebraic expressions, algebraic fractions, factoring, and the solution of linear and quadratic equations and inequalities. The interpretation and solution of verbal problems is incorporated within each skill area. Students are encouraged to develop precise and accurate habits of mathematical expression. The text for this course is Beginning Algebra with Applications. Algebra I Honors (Grades 8-9) is offered primarily to 8th grade students who completed Pre-Algebra in the 7th grade. This is an advanced course; therefore, the pace and rigor of this class will be significantly more challenging than Algebra I CP. Students will study linear, quadratic, absolute value, radical, and rational equations and inequalities, the graphing of linear and quadratic equations and inequalities, solving systems of equations and inequalities, multiplying and factoring polynomials, and simplifying exponential, radical, and rational expressions. Throughout the year, students will work extensively with word problems to develop their critical thinking skills. The text for this course is Algebra.
Caroline El-Chaar \| LinkedIn Introduction to Calculus and Vectors - taught in french Introduction to Calculus (directed to Arts and Social Sciences students) Mathematical Methods I - taught in french ... Introduction Calculus on ehow.com How to Find a Limit in Calculus \| eHow Calculus is a mathematical discipline that is based on limits. The first lessons in any introduction to calculus course concerns limits, which is the value of a ... How to Choose a Calculus Textbook \| eHow Other overall texts that are commonly used include: "Introduction to Calculus and Analysis, Volume 1" by Richard Courant and Fritz Joh as well as "Calculus, Vol. 1" by ... How to Factor in Calculus \| eHow The first lessons in any introduction to calculus course concerns limits, which... Solving Calculus Word Problems. When solving calculus word problems, it's important to ...
What's Your Number? The Key To New Resources in MathDL By Lang Moore You know that number that appears in the upper left corner of the mailing address for FOCUS? The one that you need to pay your dues online or order from the MAA Bookstore. (It has the form 000XXXXX, where each X can be any digit.) Now it is more useful than ever. It will give you access to the two new components of the MAAís Mathematical Sciences Digital Library (MathDL): MAA Reviews and Classroom Capsules and Notes. These components are available to MAA members as a privilege of membership. Below is a copy of the sign-in page for these two components. Both of these new components also are available to non-members by subscription for $25/yr. MAA Reviews, edited by Fernando Gouvêa, is the MAA's new bibliographic and reviews database, and it incorporates the MAA's Basic Library List as well. Created with the intention of replacing the old "Telegraphic Reviews" with an online service, MAA Reviews in fact goes far beyond anything the old TRs could offer. It includes a database of almost all recently-released mathematics books, a large percentage of those with reviews. Those books that have been recommended for purchase by undergraduate libraries by the MAA's Basic Library List committee are marked. The database is searchable, and the "advanced search" engine allows one to quickly find the books one wants. Classroom Capsules and Notes, edited by Wayne Roberts, provides online access to the short classroom materials that have appeared in the Associationís print journals over the years. All of us see from time to time a short article suggesting something we think we could use in the classroom: a little proof that gives unusual insight; a quick application, or connection to another area of mathematics, a question that could be used to challenge the good student. The trick is to find those items when we could actually use them. Or perhaps we have come to a point in a course where we donít recall having seen something new, but we sure wish we had. Materials in Classroom Capsules and Notes, are classified by courses, by subject, by keywords, by author, and by source, and are intended to help you quickly find that perfect enhancement to your classroom presentation.
Book Description: Points out numerical errors common in the business world. Demonstrates how to work with numbers to plan, forecast and monitor companies. Explains how to use numerical techniques to solve everyday business problems from calculating percentages and interest to evaluating competing investment opportunities.
Viewers step into the shoes of real people grappling with real-world problems whose solutions involve mathematics. This entertaining excursion allows viewers to sharpen their reasoning skills as they explore indispensable math topics such as graphing techniques, geometry, probability, and analyzing functions. Kit includes a Teacher's Resource book and student newspapers for continued learning of mathematical models. This series gives a broad overview of a basic algebra curriculum. Designed to help Algebra 1 and Algebra 2 students who need to sharpen their skills, and serves as a resource that teachers can employ to help struggling students stay up to speed. Series is hosted by professor Terry Caliste, an instructor with infectious energy and enthusiasm. The clear presentation, comprehensive activities, and Caliste's unique presentation will motivate viewers to raise their own expectations for success in math.
Intermediate Algebra : Graphs And Models - 4th edition Summary: TheBittinger Graphs and Models Serieshelps readers learn algebra by making connections between mathematical concepts and their real-world applications. Abundant applications, many of which use real data, offer students a context for learning the math. The authors use a variety of tools and techniques-including graphing calculators, multiple approaches to problem solving, and interactive features-to engage and motivate all types of learners. ...show less 2011
Queen Creek ACTGraph lines, parabolas, exponential, and logarithmic functions. (I, VIII) 7. Use the eleven field properties of the set of real numbers. (II, IV) 8. Solve quadratic equations by factoring, completing the square, and the quadratic formula. (II) 9
MATH STANDARD FOR CTE - Strand Minnesota 6-12 Academic Standards in Mathematics Selected for CTE April 14, 2007 Revision Sorted by Grade Level DRAFT Minnesota K-12 Academic Standards in Mathematics DRAFT Standards and benchmarks highlighted in yellow may be particularly applicable to CTE Courses. Page 2 of 13 Sorted by Grade April 14, 2007 DRAFT Minnesota K-12 Academic Standards in Mathematics DRAFT Recognize linear, Represent and solve problems in various contexts using linear quadratic, and quadratic functions. exponential and 9.2.2.1 For example: Write a function that represents the area of a rectangular other common garden that can be surrounded with 32 feet of fencing, and use the function functions in real- to determine the possible dimensions of such a garden if the area must be at world and least 50 square feet. mathematical situations; Represent and solve problems in various contexts using represent these 9.2.2.2 exponential functions, such as investment growth, functions with depreciation and population growth. tables, verbal Sketch graphs of linear, quadratic and exponential functions, descriptions, and translate between graphs, tables and symbolic symbols and 9.2.2.3 representations. Know how to use graphing technology to graphs; solve graph these functions. problems Express the terms in a geometric sequence recursively and by involving these giving an explicit (closed form) formula, and express the functions, and partial sums of a geometric series recursively. explain results in the original For example: A closed form formula for the terms tn in the geometric context. sequence 3, 6, 12, 24, ... is tn = 3(2)n-1, where n = 1, 2, 3, ... , and this 9.2.2.4 sequence can be expressed recursively by writing t1 = 3 and Recognize linear, tn = 2tn-1, for n  2. 9, quadratic, Another example: the partial sums sn of the series 3 + 6 + 12 + 24 + ... can 10, exponential and be expressed recursively by writing s1 = 3 and Algebra other common 11 sn = 3 + 2sn-1, for n  2. functions in real- world and mathematical situations; represent these functions with tables, verbal Recognize and solve problems that can be modeled using descriptions, finite geometric sequences and series, such as home mortgage symbols and 9.2.2.5 and other compound interest examples. Know how to use graphs; solve spreadsheets and calculators to explore geometric sequences problems and series in various contexts. involving these functions, and explain results in the original context. Sketch the graphs of common non-linear functions such as Generate f  x   x , f  x   x , f  x   1 , f(x) = x3, and translations of equivalent x 9.2.2.6 algebraic these functions, such as f  x   x 2  4 . Know how to use expressions involving graphing technology to graph these functions. Page 3 of 13 Sorted by Grade April 14, 2007 DRAFT Minnesota K-12 Academic Standards in Mathematics DRAFT polynomials and Evaluate polynomial and rational expressions and expressions radicals; use 9.2.3.1 containing radicals and absolute values at specified points in algebraic their domains. properties to evaluate expressions. Add, subtract and multiply polynomials; divide a polynomial 9.2.3.2 by a polynomial of equal or lower degree. Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two 9.2.3.3 squares. For example: 9x6 – x4 = (3x3 – x2)(3x3 + x2). Add, subtract, multiply, divide and simplify algebraic fractions. 9.2.3.4 1 x 1  2x  x 2 For example:  is equivalent to . 1 x 1 x 1 x2 Check whether a given complex number is a solution of a quadratic equation by substituting it for the variable and evaluating the expression, using arithmetic with complex numbers. 9.2.3.5 1 i 9, For example: The complex number is a solution of 2x2 – 2x + 1 = 0, 2 10, Algebra 2 since 2 1  i   2 1  i   1  i  1  i   1  0 .     11  2   2      Generate Apply the properties of positive and negative rational equivalent exponents to generate equivalent algebraic expressions, algebraic including those involving nth roots. 9.2.3.6 expressions 2  7  2 2  7 2  14 2  14 . Rules for computing 1 1 1 involving For example: polynomials and directly with radicals may also be used: 2  x  2x . radicals; use Justify steps in generating equivalent expressions by algebraic identifying the properties used. Use substitution to check the properties to equality of expressions for some particular values of the evaluate 9.2.3.7 variables; recognize that checking with substitution does not expressions. guarantee equality of expressions for all values of the Represent real- variables. Page 4 of 13 Sorted by Grade April 14, 2007 DRAFT Minnesota K-12 Academic Standards in Mathematics DRAFT world and Represent relationships in various contexts using quadratic mathematical equations and inequalities. Solve quadratic equations and situations using inequalities by appropriate methods including factoring, equations and completing the square, graphing and the quadratic formula. inequalities Find non-real complex roots when they exist. Recognize that involving linear, a particular solution may not be applicable in the original quadratic, context. Know how to use calculators, graphing utilities or 9.2.4.1 exponential, and other technology to solve quadratic equations and nth root functions. inequalities. Solve equations and inequalities For example: A diver jumps from a 20 meter platform with an upward velocity of 3 meters per second. In finding the time at which the diver hits symbolically and the surface of the water, the resulting quadratic equation has a positive and graphically. a negative solution. The negative solution should be discarded because of Interpret solutions the context. in the original Represent relationships in various contexts using equations context. involving exponential functions; solve these equations 9.2.4.2 graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. Recognize that to solve certain equations, number systems need to be extended from whole numbers to integers, from integers to rational numbers, from rational numbers to real 9.2.4.3 numbers, and from real numbers to complex numbers. In particular, non-real complex numbers are needed to solve some quadratic equations with real coefficients. Represent relationships in various contexts using systems of linear inequalities; solve them graphically. Indicate which 9.2.4.4 parts of the boundary are included in and excluded from the solution set using solid and dotted lines. Solve linear programming problems in two variables using 9.2.4.5 graphical methods. 9, 10, Algebra Represent real- 11 world and mathematical situations using Represent relationships in various contexts using absolute equations and value inequalities in two variables; solve them graphically. inequalities 9.2.4.6 For example: If a pipe is to be cut to a length of 5 meters accurate to within involving linear, a tenth of its diameter, the relationship between the length x of the pipe and quadratic, its diameter y satisfies the inequality \| x – 5\| ≤ 0.1y. exponential and nth root functions. Solve equations Page 5 of 13 Sorted by Grade April 14, 2007 DRAFT Minnesota K-12 Academic Standards in Mathematics DRAFT and inequalities Solve equations that contain radical expressions. Recognize symbolically and that extraneous solutions may arise when using symbolic graphically. methods. Interpret solutions in the original For example: The equation x  9  9 x may be solved by squaring both 9.2.4.7 context. sides to obtain x – 9 = 81x, which has the solution x   9 . However, this 80 Calculate is not a solution of the original equation, so it is an extraneous solution that measurements of should be discarded. The original equation has no solution in this case. plane and solid geometric figures; Another example: Solve 3  x 1  5 . know that physical Assess the reasonableness of a solution in its given context measurements and compare the solution to appropriate graphical or depend on the 9.2.4.8 numerical estimates; interpret a solution in the original choice of a unit context. and that they are approximations. Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as 9.3.1.1 appropriate. For example: Measure the height and radius of a cone and then use a formula to find its volume. Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface 9.3.1.2 area and volume of various figures. For example: Find the volume of a regular hexagonal prism by decomposing it into six equal triangular prisms. Understand that quantities associated with physical measurements must be assigned units; apply such units correctly in expressions, equations and problem solutions that 9.3.1.3 involve measurements; and convert between measurement systems. Calculate For example: 60 miles/hour = 60 miles/hour × 5280 feet/mile × 1 hour/3600 seconds = 88 feet/second. measurements of plane and solid geometric figures; Understand and apply the fact that the effect of a scale factor 9, know that 9.3.1.4 k on length, area and volume is to multiply each by k, k2 and 10, Algebra physical k3, respectively. 11 measurements depend on the choice of a unit Make reasonable estimates and judgments about the accuracy and that they are of values resulting from calculations involving measurements. approximations. For example: Suppose the sides of a rectangle are measured to the nearest 9.3.1.5 tenth of a centimeter at 2.6 cm and 9.8 cm. Because of measurement errors, the width could be as small as 2.55 cm or as large as 2.65 cm, with similar errors for the height. These errors affect calculations. For instance, the actual area of the rectangle could be smaller than 25 cm2 or larger than 26 cm2, even though 2.6 × 9.8 = 25.48. Page 6 of 13 Sorted by Grade April 14, 2007 DRAFT Minnesota K-12 Academic Standards in Mathematics DRAFT Understand the roles of axioms, definitions, undefined terms 9.3.2.1 and theorems in logical arguments. Accurately interpret and use words and phrases in geometric proofs such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" Construct logical 9.3.2.2 statement and its inverse, converse and contrapositive. arguments, based For example: The statement "If you don't do your homework, you can't go on axioms, to the dance" is not logically equivalent to its inverse "If you do your Geometry & definitions and homework, you can go to the dance." Measurement theorems, to prove theorems and Assess the validity of a logical argument and give other results in 9.3.2.3 counterexamples to disprove a statement. geometry. Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, 9.3.2.4 such as two-column proofs, paragraph proofs, flow charts or illustrations. For example: Prove that the sum of the interior angles of a pentagon is 540˚ using the fact that the sum of the interior angles of a triangle is 180˚. Use technology tools to examine theorems, test conjectures, perform constructions and develop mathematical reasoning 9.3.2.5 skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets. Know and apply properties of parallel and perpendicular Know and apply lines, including properties of angles formed by a transversal, properties of to solve problems and logically justify results. geometric figures 9.3.3.1 9, Geometry & to solve real- For example: Prove that the perpendicular bisector of a line segment is the 10, Measurement world and set of all points equidistant from the two endpoints, and use this fact to 11 mathematical solve problems and justify other results. problems and to Know and apply properties of angles, including logically justify corresponding, exterior, interior, vertical, complementary and results in supplementary angles, to solve problems and logically justify geometry. results. Know and apply properties of 9.3.3.2 For example: Prove that two triangles formed by a pair of intersecting lines and a pair of parallel lines (an "X" trapped between two parallel lines) are geometric figures similar. to solve real- world and mathematical Page 7 of 13 Sorted by Grade April 14, 2007 DRAFT Minnesota K-12 Academic Standards in Mathematics DRAFT problems and to Know and apply properties of equilateral, isosceles and logically justify scalene triangles to solve problems and logically justify results in 9.3.3.3 results. geometry. For example: Use the triangle inequality to prove that the perimeter of a quadrilateral is larger than the sum of the lengths of its diagonals. Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. 9.3.3.4 For example: When building a wooden frame that is supposed to have a square corner, ensure that the corner is square by measuring lengths near the corner and applying the Pythagorean Theorem. Know and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems and logically justify results. 9.3.3.5 For example: Use 30-60-90 triangles to analyze geometric figures involving equilateral triangles and hexagons. Another example: Determine exact values of the trigonometric ratios in these special triangles using relationships among the side lengths. Know and apply properties of congruent and similar figures to solve problems and logically justify results. For example: Analyze lengths and areas in a figure formed by drawing a line segment from one side of a triangle to a second side, parallel to the third side. 9.3.3.6 Another example: Determine the height of a pine tree by comparing the length of its shadow to the length of the shadow of a person of known height. Another example: When attempting to build two identical 4-sided frames, a person measured the lengths of corresponding sides and found that they matched. Can the person conclude that the shapes of the frames are congruent? Use properties of polygons—including quadrilaterals and Know and apply regular polygons—to define them, classify them, solve properties of 9.3.3.7 problems and logically justify results. geometric figures For example: Recognize that a rectangle is a special case of a trapezoid. to solve real- world and Another example: Give a concise and clear definition of a kite. 9, Know and apply properties of a circle to solve problems and Geometry & mathematical 10, logically justify results. Measurement problems and to 11 9.3.3.8 logically justify For example: Show that opposite angles of a quadrilateral inscribed in a circle are results in supplementary. geometry. Solve real-world Understand how the properties of similar right triangles allow and mathematical 9.3.4.1 the trigonometric ratios to be defined, and determine the sine, geometric cosine and tangent of an acute angle in a right triangle. Page 8 of 13 Sorted by Grade April 14, 2007 DRAFT Minnesota K-12 Academic Standards in Mathematics DRAFT problems using Apply the trigonometric ratios sine, cosine and tangent to algebraic solve problems, such as determining lengths and areas in right methods. triangles and in figures that can be decomposed into right 9.3.4.2 triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. For example: Find the area of a triangle, given the measure of one of its acute angles and the lengths of the two sides that form that angle. Use calculators, tables or other technologies in connection 9.3.4.3 with the trigonometric ratios to find angle measures in right triangles in various contexts. Use coordinate geometry to represent and analyze line 9.3.4.4 segments and polygons, including determining lengths, midpoints and slopes of line segments. Know the equation for the graph of a circle with radius r and 9.3.4.5 center (h,k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by Display and 9.3.4.6 multiples of 90˚, to solve problems involving figures on a analyze data; use coordinate grid. various measures associated with For example: If the point (3,-2) is rotated 90˚ counterclockwise about the data to draw origin, it becomes the point (2,3). 9, Geometry & conclusions, Use algebra to solve geometric problems unrelated to 10, Measurement identify trends coordinate geometry, such as solving for an unknown length 11 and describe 9.3.4.7 in a figure involving similar triangles, or using the relationships. Pythagorean Theorem to obtain a quadratic equation for a Explain the uses length in a geometric figure. of data and statistical thinking Describe a data set using data displays, such as box-and- to draw whisker plots; describe and compare data sets using summary inferences, make statistics, including measures of center, location and spread. predictions and Measures of center and location include mean, median, 9.4.1.1 justify quartile and percentile. Measures of spread include standard conclusions. deviation, range and inter-quartile range. Know how to use calculators, spreadsheets or other technology to display data and calculate summary statistics. Page 9 of 13 Sorted by Grade April 14, 2007 DRAFT Minnesota K-12 Academic Standards in Mathematics DRAFT Analyze the effects on summary statistics of changes in data sets. For example: Understand how inserting or deleting a data point may affect 9.4.1.2 the mean and standard deviation. Another example: Understand how the median and interquartile range are affected when the entire data set is transformed by adding a constant to each data value or multiplying each data value by a constant. Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine 9.4.1.3 regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. Use the mean and standard deviation of a data set to fit it to a normal distribution (bell-shaped curve) and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets and tables to estimate areas under the normal curve. 9.4.1.4 For example: After performing several measurements of some attribute of an irregular physical object, it is appropriate to fit the data to a normal distribution and draw conclusions about measurement error. Another example: When data involving two very different populations is combined, the resulting histogram may show two distinct peaks, and fitting the data to a normal distribution is not appropriate. Evaluate reports based on data published in the media by identifying the source of the data, the design of the study, and the way the data are analyzed and displayed. Show how graphs and data can be distorted to support different points of 9.4.2.1 view. Know how to use spreadsheet tables and graphs or graphing technology to recognize and analyze distortions in data displays. For example: Shifting data on the vertical axis can make relative changes appear deceptively large. Identify and explain misleading uses of data; recognize when Calculate 9.4.2.2 arguments based on data confuse correlation and causation. probabilities and 9, Data apply probability 10, Analysis & concepts to solve 11 Probability real-world and mathematical Explain the impact of sampling methods, bias and the problems. 9.4.2.3 phrasing of questions asked during data collection. Page 10 of 13 Sorted by Grade April 14, 2007 DRAFT Minnesota K-12 Academic Standards in Mathematics DRAFT Select and apply counting procedures, such as the multiplication and addition principles and tree diagrams, to determine the size of a sample space (the number of possible outcomes) and to calculate probabilities. 9.4.3.1 For example: If one girl and one boy are picked at random from a class with 20 girls and 15 boys, there are 20 × 15 = 300 different possibilities, so the probability that a particular girl is chosen together with a particular boy 1 is . 300 Calculate experimental probabilities by performing 9.4.3.2 simulations or experiments involving a probability model and using relative frequencies of outcomes. Understand that the Law of Large Numbers expresses a relationship between the probabilities in a probability model 9.4.3.3 and the experimental probabilities found by performing simulations or experiments involving the model. Use random numbers generated by a calculator or a spreadsheet, or taken from a table, to perform probability simulations and to introduce fairness into decision making. 9.4.3.4 For example: If a group of students needs to fairly select one of its members to lead a discussion, they can use a random number to determine the selection. Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. 9.4.3.5 For example: The probability of tossing at least one head when flipping a fair coin three times can be calculated by looking at the complement of this event (flipping three tails in a row). Describe the concepts of intersections, unions and Calculate 9, Data complements using Venn diagrams. Understand the probabilities and 10, Analysis & 9.4.3.6 relationships between these concepts and the words AND, apply probability 11 Probability OR, NOT, as used in computerized searches and concepts to solve spreadsheets. real-world and Page 11 of 13 Sorted by Grade April 14, 2007 DRAFT Minnesota K-12 Academic Standards in Mathematics DRAFT mathematical Understand and use simple probability formulas involving problems. intersections, unions and complements of events. For example: If the probability of an event is p, then the probability of the 9.4.3.7 complement of an event is 1 – p; the probability of the intersection of two independent events is the product of their probabilities. Another example: The probability of the union of two events equals the sum of the probabilities of the two individual events minus the probability of the intersection of the events. Apply probability concepts to real-world situations to make informed decisions. For example: Explain why a hockey coach might decide near the end of the 9.4.3.8 game to pull the goalie to add another forward position player if the team is behind. Another example: Consider the role that probabilities play in health care decisions, such as deciding between having eye surgery and wearing glasses. Use the relationship between conditional probabilities and relative frequencies in contingency tables. 9.4.3.9 For example: A table that displays percentages relating gender (male or female) and handedness (right-handed or left-handed) can be used to determine the conditional probability of being left-handed, given that the gender is male. 9, Data 10, Analysis & 11 Probability Page 12 of 13 Sorted by Grade April 14, 2007 DRAFT Minnesota K-12 Academic Standards in Mathematics DRAFT Page 13 of 13 Sorted by Grade April 14, 2007
Looking for support with your Algebra course, then the Algebra Buster is your solution. This software explains all the steps to the problems you enter. Jacob Matheson, FL This is excellent tutoring software, it really helped get my grades up, and it is so easy even a total dummy like me could do it. Stephanie Cummings, AZ This is great, finishing homework much faster31 : algebra programs easy gears worksheet how to do cubed root on calculator adding and subtractng fractions like denominators ged math practice sheets convert 56% to a fraction ti-89 polar graph tutorial McDougal Littell chemistry procedures for lab 20 calculator for solving quadratic equations by a+bi adding and subtracting mixed numbers worksheet graphing equations using limits math unit 5 divide whole numbers and decimals worksheet what is the difference between equivalent equations and simplification
Relations, Functions and Linear Equations: Study the definition of relations and functions. Use of mapping, tables, ordered sets and graph to represent relations and functions. Learn how to apply the vertical test to identify a function. Determine the difference between discrete vs. continues functions. Definition of linear equations and the Standard Form. Learning About Slope: Slope formula (run vs. rise and two points), falling to the right, horizontal, vertical, negative and positive cases for the slope. Slope of parallel and perpendicular lines. Working With Systems of Linear Equations: Slope intercept form and point-slope form problems that involve a point and slope, two points, etc. Solving systems of two variable linear equations by substitution, linear combinations and graphing. Introduction to special functions (step function, constant function, identity function and absolute value function)
◊Business Technology Pre-Algebra Mathematics Prepares students who want to strengthen computational and problem-solving skills before proceeding to an algebra course. Reviews arithmetic and measurements (both metric and American). Teaches the concept of variables, operations involving signed numbers, simplifying algebraic expressions, solving equations and inequalities in one variable, solving simple formulas, ratio and proportion, and solving application problem using equations. Prereq: MATH 1 or placement at MATH 22, and ENG 19 with grade C or better or placement at least ENG 22; or consent.
The Math Curriculum is standard-based, integrated and aligned with the Washington State Essential Academic Learning Requirements. The Common Core Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students.These practices rest on important "processes and proficiencies" with longstanding importance in mathematics education. The first of these are problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency
Catalog of MAA Publications 2011 Annual : Page 8 NEW Lie Groups A Problem-Oriented Introduction via Matrix Groups Harriet Pollatsek ■ MAA Textbooks Can be used as supplementary reading in a linear algebra course or as a primary text in a "bridge" course that helps students make the transition to courses that emphasize definition and proofs, as well as for an upper level elective. The work of the Norwegian mathematician So-phus Lie extends ideas of symmetry and leads to many applications in mathematics and physics. Ordinarily, the study of the "objects" in Lie's theory (Lie groups and Lie algebras) requires exten-sive mathematical prerequisites beyond the reach of the typical undergrad-uate. By restricting to the special case of matrix Lie groups and relying on ideas from multivariable calculus and linear algebra, this lovely and im-portant material becomes accessible even to college sophomores. Working with Lie's ideas fosters an appreciation of the unity of mathematics and the sometimes surprising ways in which mathematics provides a language to describe and understand the physical world. This is the only book in the undergraduate curriculum to bring this material to students so early in their mathematical careers. Geometric Transformations IV Circular Transformations I. M. Yaglom Translated by Abe Shenitzer ■ Anneli Lax NML The familiar plane geometry of high school— figures composed of lines and circles—takes on a new life when viewed as the study of properties that are preserved by special groups of transformations. No longer is there a single, universal geometry: different sets of transformations of the plane correspond to intriguing, disparate geometries. This book is the concluding Part IV of Geometric Transformations , but it can be studied independently of Parts I, II, and III, which appeared in this series as Volumes 8, 21, and 24. Part I treats the geometry of rigid motions of the plane (isometries); Part II treats the geometry of shape-preserving transformations of the plane (similarities); Part III treats the geometry of transformations of the plane that map lines to lines (affine and projective transformations) and introduces the Klein model of non-Euclidean geometry. The present Part IV develops the geometry of transformations of the plane that map circles to circles (conformal or anallagmatic geometry). The notion of inversion, or reflection in a circle, is the key tool employed. Applications include ruler-and-compass constructions and the Poincaré model of hyper-bolic geometry. The straightforward, direct presentation assumes only some background in high school geometry and trigonometry. Numerous exercises lead the reader to a mastery of the methods and concepts. The second half of the book contains detailed solutions of all the problems. 164 pp., 2009 List: $63.95 ISBN: 978-0-88385-759-5 MAA Member: $51.95 Hardbound Catalog Code: LIG/YD11 Visual Group Theory Nathan Carter ■ Classroom Resource Materials Could serve as a text in abstract algebra/ group theory at the undergraduate level, or as supplementary reading at the graduate level. 296 pp., 2009 List: $46.95 ISBN: 978-0-88385-648-2 MAA Member: $36.95 Paperbound Catalog Code: NML-44/YD11 Over 300 illustrations printed in full color. In a New York Times article, Steven Strogatz of Cornell University calls Visual Group Theory a "terrific new book." He describes the book as "one of the best introductions to group theory—or to any branch of higher math—I've ever read." The more than 300 illustrations in Visual Group Theory bring groups, sub-groups, homomorphisms, products, and quotients into clear view. Every topic and theorem is accompanied with a visual demonstration of its mean-ing and import, from the basics of groups and subgroups through advanced structural concepts such as semidirect products and Sylow theory. Although the book stands on its own, the free software Group Explorer makes an excellent companion. It enables the reader to interact visually with groups, including asking questions, creating subgroups, defining homomorphisms, and saving visualizations for use in other media. It is open source software available for Windows, Macintosh, and Unix systems from Flatland Edwin Abbott Notes and commentary by William F. Lindgren & Thomas F. Banchoff ■ Spectrum Flatland , Edwin Abbott's story of a two-dimen-sional universe as told by one of its inhabitants who is introduced to the mysteries of three-dimensional space, has enjoyed an enduring popularity from the time of its publication in 1884. This fully annotated edition enables the modern-day reader to under-stand and appreciate the many "dimensions" of this classic satire. Mathe-matical notes and illustrations enhance the usefulness of Flatland as an elementary introduction to higher-dimensional geometry. Historical notes show connections to late-Victorian England and to classical Greece. Citations from Abbott's other writings, as well as the works of Plato and Aristotle, serve to interpret the text. Commentary on language and literary style in-cludes numerous definitions of obscure words. An appendix gives a compre-hensive account of the life and work of Flatland 's remarkable author. 334 pp., 2009 List: $71.95 ISBN: 978-0-88385-757-1 MAA Member: $57.50 Hardbound Catalog Code: VGT/YD11 296 pp., 2010 List: $14.99 ISBN: 978-0-52175-994-6 Paperbound Catalog Code: FTL/YD11 5 8 To Order : Call 1.800.331.1622 or Online at
what is the different between spec math and math? please anyone explain.. thanks! spec math is totally different from math(or the full term : mathematical studies) spec math is totally much more difficult that math studies. cos in math studies you just learn mostly on some algebra, differentiation and intergration.. and questions are mostly on these stuff.. where as for spec math, the questions are normally much more on proving equations like " show that this = that ", then you'll have to apply some theories or laws to prove that this = that. spec math is normally taken by engineering students. though some medic students does take it as well. say in AUSMAT 17, out of 300 students in our batch, about 70 were medic students. and out of this 70 medic students less than 5 did actually take spec mathahhh... okay. okay. looks like calculus is gonna b a tough subject, huh? how bout the other question? the difference between applicable math and discrete math is??ahhh... okay. okay. looks like calculus is gonna b a tough subject, huh? how bout the other question? the difference between applicable math and discrete math is?? yeah.. its quite true alright... what ever you study for your matriculation.. it might seem related to what you study in form 5.. in fact it is related.. but when you really get into it.. its a whole new level for you and definitely not as easy as it seems in form 5
0738609277 9780738609270 California Star Mathematics, Grades 8-9: Every eighth and ninth grade student in California must participate in the STAR program Are you ready for the STAR Mathematics Exam? REA's California STAR Grades 8 & 9 Mathematics test prep helps you sharpen your skills and pass the exam! Fully aligned with the learning standards of the California Department of Education, this second edition of our popular test prep provides the up-to-date instruction and practice that eighth and ninth grade students need to improve their math skills and pass this important state-required exam. The comprehensive review features student-friendly, easy-to-follow lessons and examples that reinforce the key concepts tested on the STAR, including: ArithmeticAlgebraGeometryData AnalysisStatisticsWord ProblemsFocused lessons explain math concepts in easy-to-understand language that's suitable for eighth and ninth grade students at any learning level. Our tutorials and targeted drills increase comprehension while enhancing your math skills. Color icons and graphics throughout the book highlight practice problems, charts, and figures. The book contains four diagnostic tests that are perfect for classroom quizzes, homework, or extra study. A full-length practice exam lets you test your knowledge and reinforces what you've learned. The practice test comes complete with detailed explanations of answers, allowing you to focus on areas in need of further study. REA's test-taking tips and strategies give you an added boost of confidence so you can succeed on the exam. Whether used in a classroom, at home for self-study, or as a textbook supplement, teachers, parents, and students will consider this book a "must-have" prep for the STAR. REA test preps have proven to be the extra support students need to pass their challenging state-required tests. Our comprehensive test preps are teacher-recommended and written by experienced educators. «Show less Rent California Star Mathematics, Grades 8-9 2nd Edition today, or search our site for other Hearne Tests
Linear algebra forms the basis for much of modern mathematics—theoretical, applied, and computational. Finite-Dimensional Linear Algebra provides a solid foundation for the study of advanced mathematics and discusses applications of linear algebra to such diverse areas as combinatorics, … Comprehensive Coverage of the New, Easy-to-Learn C# Although C, C++, Java, and Fortran are well-established programming languages, the relatively new C# is much easier to use for solving complex scientific and engineering problems. Numerical Methods, Algorithms and Tools in C# presents a broad … This book provides a comprehensive overview of the computational physics for nanoscience and nanotechnology. Based on MATLAB and the C++ distributed computing paradigm, the book gives instructive explanations of the underlying physics for mesoscopic systems with many listed programs that readily … Unlike … Although the Trefftz finite element method (FEM) has become a powerful computational tool in the analysis of plane elasticity, thin and thick plate bending, Poisson's equation, heat conduction, and piezoelectric materials, there are few books that offer a comprehensive computer programming … Divided into three main parts, the book guides the reader to an understanding of the basic concepts in this fascinating field of research. Part 1 introduces you to the fundamental concepts of simulation. It examines one-dimensional electrostatic codes and electromagnetic codes, and describes the … The use of computer algebra systems in science and engineering has grown rapidly as more people realize their potential to solve tedious and extensive mathematical problems. REDUCE for Physicists provides a comprehensive introduction to one of the most widely available and simple to use computer …
TI-Nspire Strategies: Algebra Resource Book & CD This book move students from concrete understanding of mathematical concepts through the abstract comprehension level, to real-life application, while at the same time allowing students to develop skill in the use of the handheld. As a result of this technology, teachers are able to engage students more effectively by addressing different learning styles & developing understanding that leads to higher-level thinking.
Michael C. Walker Math Center Students having difficulty with mathematical skills are encouraged to make use of the free peer tutoring services offered in the Mathematics Center. Students are also encouraged to utilize the solution manuals and alternative texts that are provided in the Center for their benefit. The Center is a hub of activity. The Math Club uses this setting as their home base. The majors enjoy time with each other, either discussing real work or real life, and the tutors welcome all others who would like to know more. Majors and non-majors alike use it as a valuable learning resource as well as a gathering place.
Algebra Help - Algebra Answers Algebra Homework Help and Answers Hotmath explains the odd-numbered homework problems for algebra textbooks used in middle school, high school, and college. We show step-by-step algebra answers to the homework problems. Click below for a list of the algebra textbooks we cover plus free sample solutions:
New developments in many applications, such as weather forecasting, airplane design, tomographic problems, analysis of the stability of structures, design of chips and other electrical circuits, etc, rely on numerical simulations. Such simulations require the numerical solution of linear systems or of eigenvalue problems. The matrices involved are sparse and high dimensional (1 billion is not acceptional). The solution of these linear problems are normally by far the most time-consuming part of the whole simulation. Therefore, the development of new solution algorithms is extremely important and forms a very active area of research. The course will give an overview of the modern solution algorithms for linear systems and eigenvalue problems. Modern approaches rely on schemes that improve approximate solutions iteratively. The course will start with a review of basic concepts from linear algebra, after which solution methods for dense systems (LU, QR and Choleski decomposition) will be discussed. Next, the basic ideas for iterative solution methods of sparse systems will be explained, which will lead to the main topic of the course: modern Krylov subspace methods. The main ideas of these methods will be explained and how they lead to efficient solvers. Solution algorithms for linear systems that will be discussed include CG, GMRES, CGS, Bi-CGSTAB, Bi-CGSTAB(l) and IDR(s). Furthermore several preconditioning and deflation techniques will be explained. For large scale eigenvalue problems the Lanczos methods, Arnoldi's method and the Jacobi-Davidson method will be treated. Organization Fourteen lectures, each consisting of instruction and theoretical and practical assignments. The practical assignments require programming in MATLAB. Examination Quiz, homework assignments and a final project assignment. Prerequisites Good knowledge of linear algebra and some experience in programming in MATLAB.
Mathematics for EconomicsWere you looking for the book with access to MyMathLab Global? This product is the book alone, and does NOT come with access to MyMathLab Global. Buy Mathematics for Economics and Business with MyMathLab Global access card, 7/e (ISBN 9780273788492) if you need access to the MyLab as well, and save money on this brilliant resource. With its friendly and informal style, this market leading text breaks down topics into short sections making learning each new technique seem less daunting. With plenty of ... MOREpractice problems, it provides opportunities to stop and check understanding and allows students to learn at their own pace Mathematics for Economics and Business with MyMathLab Global access card, 7/e (ISBN 9780273788492). Alternatively, buy access online at If you want to increase your confidence in mathematics then look no further. This book breaks topics down into short sections making each new technique you learn seem less daunting. Stop and check your understanding along the way, working through the practice problems, and learn at your own pace. Learning online with MyMathLab Go to - your gateway to all the online resources for this book: MyMathLab provides you with the opportunity for unlimited practice, guided solutions with tips and hints to help you solve challenging questions, an interactive e-book and a personalised study plan to help focus your revision efforts on the topics you need most support with. If you have purchased this text as part of a pack then you can gain access to MyMathLab by following the instructions to register on the access code included on the enclosed access card. If you've purchased this text on its own then you can purchase access online at See the Getting Started with MyMathLab in 'Guided Tour' area of this text for more details. Preface Guided tour of the textbook Guided tour of the online resources Introduction: Getting Started Notes for students: how to use this book CHAPTER 1 Linear Equations 1.1 Introduction to algebra 1.2 Further algebra 1.3 Graphs of linear equations 1.4 Algebraic solution of simultaneous linear equations 1.5 Supply and demand analysis 1.6 Transposition of formulae 1.7 National income determination
Content on this page requires a newer version of Adobe Flash Player. Welcome to Focus on Algebra. This resource site was developed to support math educators and educational leaders as they develop students' algebraic thinking, utilize formative assessment, and provide intervention support in an effort to have all students reach success in algebra. Watch the introduction video and then select your destination from the menu below.
The Baldwin Math Department recommends summer practice to maintain math skills. Summer Math Skills Sharpenerbooklets are available through the website The cost per book is $20. Each booklet contains 30 pages; content is written in accordance with national standards. The overall design of each booklet is to provide 15 – 30 minutes of skills practice per page. Three pages per week is a reasonable pace. If your daughter is entering 6thgrade in the fall, we recommend the 6thgrade math booklet. If your daughter is entering 7thgrade in the fall, we recommend the 7thgrade math booklet. If your daughter is entering 8thgrade in the fall, we recommend the Prealgebra booklet. If your daughter is entering 9thgrade in the fall but hasn't yet had Algebra 1, we recommend the Prealgebra booklet. Finally, if your daughter is entering 9thgrade or above and has had Algebra 1, we recommend the Algebra 1 booklet.
Study Plan, Notes and Exercises This MEI resource for OCR FP1 provides information on complex numbers, equations and geometric representation. Topics include sections on polynomial equations with complex roots, representing complex numbers geometrically and roots of complex numbers. There are also notes which cover the factor theorem, a study plan and crucial points to learn. Additional exercises for students to complete, with worked answers, as well as a multiple choice test including solutions are also available
Algebra 2 is a lot harder than it used to be. It's also more important than it used to be because algebra 2 concepts are included on the new SAT. Sadly, most of today's students have very limited vocabularies.
Features Provides case studies that illustrate how to assess the complexity of a problem Figure slides available upon qualifying course adoption Summary gives a practical treatment of algorithmic complexity and guides readers in solving algorithmic problems. Divided into three parts, the book offers a comprehensive set of problems with solutions as well as in-depth case studies that demonstrate how to assess the complexity of a new problem. Part I helps readers understand the main design principles and design efficient algorithms. Part III supplies readers with tools and techniques to evaluate problem complexity, including how to determine which instances are polynomial and which are NP-hard. Drawing on the authors' classroom-tested material, this text takes readers step by step through the concepts and methods for analyzing algorithmic complexity. Through many problems and detailed examples, readers can investigate polynomial-time algorithms and NP-completeness and beyond. Author Bio(s) Editorial Reviews "This book is unique among texts on algorithmics in its emphasis on how to 'think algorithmically' rather than just how to solve specific (classes of) algorithmic problems. The authors skillfully engage the reader in a journey of algorithmic self-discovery as they cover a broad spectrum of issues, from the very basic (computing powers, coin changing) through the quite advanced (NP-completeness, polynomial-time approximation schemes). The authors emphasize algorithmic topics that have proven useful in 'applied' situations … . I shall be very happy to have this text on my bookshelf as a reference on methods as well as results." —Arnold L. Rosenberg, Research Professor, Northeastern University, and Distinguished University Professor Emeritus, University of Massachusetts Amherst "This book presents a well-balanced approach to theory and algorithms and introduces difficult concepts using rich motivating examples. It demonstrates the applicability of fundamental principles and analysis techniques to practical problems facing computer scientists and engineers. You do not have to be a theoretician to enjoy and learn from this book." —Rami Melhem, Professor of Computer Science, University of Pittsburgh
The new 4th Edition 7th and 8th level math PACEs will strengthen your students' current math skills while preparing for future math studies. Beginning with a comprehensive review of basic math concepts, students will be introduced to prealgebra, pregeometry, and consumer math that will fully equip them for high school math. Students working in 4th Edition 6th level math should continue with the new 7th level PACEs without being rediagnosed and then proceed into 8th level PACEs. Students working in 7th level Intermediate Math must be rediagnosed. When rediagnosing students, it is important that they complete all gap PACES before proceeding in 8th level PACEs. Procedures Manual Reminder: Use of calculators should be permitted starting in 7th level and thereafter, unless otherwise noted in the PACE Goal page, but use should be permitted only after the student has successfully demonstrated competence in manual computational skills. Note: Existing Intermediate (3rd Edition) Math PACEs 1085–1096 (item #6085–#6096) and Keys (#6285, #6288, #6291, and #6294) will only be available until January 2013. Thank you for training your students to be perseverant in meeting their daily goals. Perseverant—Withstanding stress (the attacks of time and circumstance) to accomplish God's best. And let us not be weary in well doing: for in due season we shall reap, if we faint not. Galatians 6:9
Recently Viewed Princeton Companion To Mathematics Synopsis Includes entries, which introduce basic mathematical tools and vocabulary; trace the development of modern mathematics; define essential terms and concepts and put them in context; explain core ideas in major areas of mathematics; and describe the achievements of scores of famous mathematicians. Details Country of Origin : UNITED STATES Editor : Gowers, Timothy Illustrations : Black-And-White Illustrations Throughout \| Cross-References, Bibliographies, Index Number Of Pages : 1008 Year of Publication : 2008
What are functions ? From an introduction of the basic concepts of functions to more advanced functions met in economics, engineering and the sciences, these topics provide an excellent foundation for undergraduate study. Composition of functions Complicated functions can be built up from simple functions by using the process of composition, where the output of one function becomes the input of another. It is also sometimes necessary to decompose a function into two simpler functions. Video tutorial 11 mins.
Participants will explore selected functions from multiple perspectives. Functions of interest may include polynomials and trigonometric functions, but also matrix functions and geometric transformations. One possible topic would be the use of computer algebra systems to work with functions, and how these can support or hinder learning. For this topic teachers could study relevant research, design and evaluate lessons based on this study, or discuss implications for their own work with students
MATH E-3 Quantitative Reasoning: Practical Math This course reviews basic arithmetical procedures and their use in everyday mathematics. It also includes an introduction to basic statistics covering such topics as the interpretation of numerical data, graph reading, hypothesis testing, and simple linear regression. No previous knowledge of these tools is assumed. Recommendations for calculators are made during the first class. Prerequisite: a willingness to (re)discover math and to use a calculator. (4 credits)
Five Important Exponential Models by This file has 5 tabs that illustrate important exponential models.In each tab the student may use the sliders in the lower left to explore the effects of the parameters in the model.The yellow high-lighted point on the x-axis may also be dragged to alter the function values in the upper right of the screen. Illustrations for AP Statistics Exploring Rational Funtions by This file has 5 tabs that allow students to explore the reciprocal functions of 2nd and 3rd degree equations, simple rational functions, more complex rational functions with horizontal asymptotes and finally rational functions with slant asymptotes. Students can manipulate the roots of the numerator and denominator functions and see how these affect the behavior of a rational function. Roots Control Local Behavior by This file allows students to drag the roots of polynomials along the x axis and examine both the global and local behavior of the graph. This helps students to see how the multiplicity of a root affects the local behavior of the graph. There are tabs for second, third, fourth, and fifth degree polynomials. Three Formats for Parabolas Trasforming Functions by This file allows students to examine the effects of altering four parameters a, b, c, and d. These parameters create both rigid and non-rigid transformations of any function, f(x). There are six tabs that can be accessed at the bottom of the screen.
principal aim of this book is to introduce university level mathematics — both algebra and calculus. The text is suitable for first and second year students. It treats the material in depth, and thus can also be of interest to beginning graduate students. New concepts are motivated before being introduced through rigorous definitions. All theorems are proved and great care is taken over the logical structure of the material presented. To facilitate understanding, a large number of diagrams are included. Most of the material is presented in the traditional way, but an innovative approach is taken with emphasis on the use of Maple and in presenting a modern theory of integration. To help readers with their own use of this software, a list of Maple commands employed in the book is provided. The book advocates the use of computers in mathematics in general, and in pure mathematics in particular. It makes the point that results need not be correct just because they come from the computer. A careful and critical approach to using computer algebra systems persists throughout the text.
MA141 - College Trigonometry Course Description: MA141 College Trigonometry: A consideration of those topics in trigonometry necessary for the calculus. Topics include: circular functions, identites, special trigometric formulas, solving triangles, polar coordinates, vectors, and conic sections. 3:0:3 Prerequisite: MA135, or a high school or transfer course equivalent to MA135, or ACT math score greater than or equal to 26, or an SAT math score greater than or equal to 560, or a COMPASS score greater than or equal to 46 in the College Algebra placement domain. @ (From catalog 2011-2012)
Courses Course Details MATH 096 Intermediate Algebra and Geometry 5 hours lecture, 5 units Letter Grade or Pass/No Pass Option Description: Intermediate algebra and geometry is the second of a two-course integrated sequence in algebra and geometry. This course covers systems of equations and inequalities, radical and quadratic equations, quadratic functions and their graphs, complex numbers, nonlinear inequalities, exponential and logarithmic functions, conic sections, sequences and series, and solid geometry. The course also includes application problems involving these topics. This course is intended for students preparing for transfer-level mathematics courses.
Maths a subject that tests your Problem-solving approach, it uses different numbers, variables, functions and theorems to reach to the solution of any problem. BITS iText in ActionWriting a book forManning Publications Co.What is iText?• First PDF library: rugPdf (1998)– One Christmas holiday of study– Developed in six weeks time– Not user-friendly: for PDF experts only• 2000: I want my own project!– Rebuild a PDF library from scratch– For people who don't know anything about PDF– iText hosted at lowagie.comorialee tutFr2004: First contact• November 2004: offer to write a book forO'Reilly• April 2005: offer to write a book forManningPreparing a book• First contact: Publisher's Assistant• After evaluation: Acquisition Manager:helps writing the Book Proposal– Short Summary– Detailed Table of Contents– Marketing Overview• Industry reviewBook contract• 5 deadlines– Chapter 1– 1/3 of the manuscript– 2/3 of the manuscript– 3/3 of the manuscript– Final manuscript• "Author Launch"Writing a book• Development Manager assigns:– Developmental Editor• Helps the writer "developing" his book• Obey the Manning Style Guide!• Review Manager• Coordinates the review processProduction• Production Manager assigns:– Copy Editor– Technical Editor– Lay-out– Proof Reader• Cover Designer• Marketing Manager: go or no-go• Back cover: written by the Publisher himselfTimingFirst Edition:• May – July 2005: preparing the book• September 2005 – February 2006: writing• March – November 2006: productionSecond Edition:• August – September 2009: preparing• October 2009 – April 2010: writing• May – August 2010: productionThe result
Paperback Click on the Google Preview image above to read some pages of this book! Nelson QMaths for the Australian Curriculum 7 ï½ï½ï½ 10 is a brand new series that has been developed to support teachers implementing the Australian Mathematics Curriculum for Years 7 ï½ï½ï½ 10 students in Queensland. A comprehensive range of resources are available in printed form and in digital form on NelsonNet to support the Nelson QMaths series. This is the Teacher's Edition of the Year 9 student textbook. It contains the same content as the student book with additional page-by-page wraparound information to assist teachers with lesson planning and instruction. It includes suggests for integrating key aspects of the curriculum (capabilities, proficiencies, technology, and cross-curriculum priorities) into the teaching of each topic.
97807637149 Analysis, Revised Edition (Jones and Bartlett Books in Mathematics) The Way Of Analysis Gives A Thorough Account Of Real Analysis In One Or Several Variables, From The Construction Of The Real Number System To An Introduction Of The Lebesgue Integral. The Text Provides Proofs Of All Main Results, As Well As Motivations, Examples, Applications, Exercises, And Formal Chapter Summaries. Additionally, There Are Three Chapters On Application Of Analysis, Ordinary Differential Equations, Fourier Series, And Curves And Surfaces To Show How The Techniques Of Analysis Are Used In Concrete Settings
Variation_1 Self-learning modular algebra program on understanding and solving the different types of variations including direct, inverse, variation as sum of two parts, joint and combined variations using step-by-step approach. Only inverse variation is enabled. Try it!
"We LOVE Uncle Buck's clear explanation of Algebra. My husband has even commented that he would like to join us for learning advanced math from Dana Mosely when we reach that stage in our son's education. These teaching DVDs are a real treasure and will remain in our family library for the next generation as well (our grandkids). :)" With gratitude, Beverly Rohrer "Our two oldest children used Chalkdust Algebra several years ago and now our 8th grader is using the Prealgebra and they all say there is no other math curriculum they like as good as Chalkdust. When our daughter began 9th grade in a private school she took Algebra 2. She said everything she learned in Algebra 2 she had already learned in Chalkdust Algebra 1, which she had taken the year before at home. Thanks again." The text for this course is Algebra 1 by Larson ad Hostetler in a special edition published for Chalk Dust. This is a traditional course targeted toward the average student. Mosely's thorough presentations plus the combination of video instruction with textbook reinforcement should make it easy for most students to master algebra while working independently through the course. In addition, the solution guide will help when both students and parents are stumped. While this text does not incorporate geometry instruction (as does Saxon) it does include algebraic applications in geometry. (Chalk Dust offers a separate Geometry course, as do most publishers.) The Algebra 1 course has lessons and exercises for using a graphing calculator. In addition, the book's appendix adds sections on graphing calculators, geometry, and statistics, but the appendix sections are not included on video. (An Index of Applications is also included in the appendix.) "I have two children--one who graduated from high school in 2003 and a sophmore in high school. I have used Chalkdust's Pre-algebra, Algebra, Geometry and Algebra II. My oldest daughter is taking Calculus in college and is scoring at 98%. I attribute her success to her excellent foundation in math from Chalkdust! Thank you." Karen Potter
Share this knowledge with your friends! Very well done. I'll admit I was a little confused by the initial lecture on Linear Systems, but the examples themselves were very informative and helpful. 0 answers Post by Jason Mannionon October 4, 2011 I subscribed to this site because I was having great difficulties in my university Linear Algebra class. So I have only watched the "Linear Systems" lecture, and from that I can conclude a few things: 1)Dr. Hovasapian actually has the ability to organize a course, 2) He knows how to present material in a educational manner, and 3) in the one video I learned more than I did in several weeks of my university lectures. (the problem is simply that my professor lacks any teaching abilities, and cannot organize the material. We started our course with complex numbers, and then went straight into vectors and subspaces, without any explanation as to what a linear system was, how to solve one, or even how to do basic matrix arithmetic! In fact, we never see any matrices in class!) 0 answers Post by Arthur Booksteinon October 12, 2011 Very well explained. 0 answers Post by Senghuot Limon December 18, 2011 Einstein? 0 answers Post by thomas kotchon December 18, 2011 He is great! 0 answers Post by amir szeinbergon February 18, 2012 I have a problem re-entering the lecture in the middle, say in the 20th minute, and I have to strart from the begining. Is there any thing I could do about it? Thanks in advance, Amir I will be taking this class soon. Man it look hard :( I will be fine, I have Educator now! ^^ 0 answers Post by Maimouna Loucheon June 17, 2012 Thanks I get it now, it looked scary for nothing. 0 answers Post by Nik Googoolion August 24, 2012 very bad hand writing 0 answers Post by Nik Googoolion August 24, 2012 he is so clever, but he never be a good teacher,,,, 1 answer Last reply by: Professor Hovasapian Fri Sep 21, 2012 2:34 PM Post by Erdem Balikcion September 20, 2012 Very well done! Clear and smooth!! 2 answers Last reply by: Rob Lee Thu Sep 27, 2012 6:18 PM Post by robert leeon September 26, 2012 Question: at 36:02. I am slightly confused with your graph, I understand the no solution case, where the lines are parallel and never meet. The one solution case meets at one point (in other words if it was three or more lines, they should all intersect at the same point then?) Now the infinite case, I do not understand at all, what does it mean when the line is on top of another line? Doesn't this just mean that they meet at one point? So it would be just like the one solution case then... How is this possible? In my imagination, an infinite solution would be more like a graph of a sin and cos equation, cause then they would intersect at multiple points, but then this would not be linear?? Can you please clarify? Thank you. 3 answers Last reply by: Aniket Dhawan Thu Oct 11, 2012 5:09 AM Post by Suhaib Hasanon October 4, 2012 Your comment about induction and math was great. 1 answer Last reply by: Professor Hovasapian Thu Oct 11, 2012 3:29 PM Post by Aniket Dhawanon October 11, 2012 You are a great teacher. I really liked your explanations,they helped me a lot. Thankyou professor 1 answer Last reply by: Professor Hovasapian Fri Apr 12, 2013 4:33 PM Post by Rishabh Jainon April 12 at 05:40:06 AM I love your lecture SIR !!! AMAZINGGGGGGGGG !!!!!!!!!! Linear Systems
Undergraduate Programs in Mathematics & Statistics at Lehigh University Entry level math courses There are three calculus sequences. Calculus 21, 22, 23 is the largest, taken by most science and engineering students. Advanced placement can be obtained either through the AP test of the College Board or through a test administered by our department. Honors Calculus 31, 32, 33 parallels 21, 22, 23, but with more depth and rigor. It is geared toward students with SAT Math score over 700, although it is open to all interested students. Survey of Calculus, 51, 52, are somewhat less in-depth. Most business majors will take 21. Most students in biological or earth sciences will take 51 and 52. For students who need to take Calculus 21-23, but have a weak background in pre-calculus material, there is a 2-semester course, Math 75 and 76, which can be substituted for the 1-semester course Math 21. Math 75 and 76 contain a good bit of review of pre-calculus material along with the topics of Math 21. Students who complete 75 and 76 will be prepared to take Math 22. Another option is Math 0, Preparation for Calculus, offered during the fall. It counts as 2 credits on your current roster and your GPA, but the credits do not count toward graduation. Math 0 will be taken primarily by students who need to take Math 51, but fail the Calculus Readiness Exam. Students in the arts, humanities, and social sciences are required to take at least one semester of mathematics, but it need not be calculus. Basic Statistics 12 is the math course recommended for social science students. It is a 4-credit course and is offered every semester. Another non-calculus-based course is Math 5, Introduction to Mathematical Thought, offered in the spring. Topics in Math 5 vary from year to year. The math department particpates in the College of Arts and Sciences Freshman Seminar Program and offers a freshman seminar each fall semester. Topics for the freshman seminar vary from year to year. B.A. in Mathematics This is a math major in a liberal arts tradition. It prepares students for a variety of careers in government, industry, and education. The required major courses are Math 21,22,23 or 31,32,33........Calculus or Honors Calculus Math 163.......Introductory Seminar Math 12 or 2312 more math courses B.S. General Mathematics Option This is the recommended program for students who wish to go on for a Ph.D. in Mathematics4 more math courses Two CSC courses or one CSC course and Engr 1 B.S. Applied Mathematics Option This provides a broad background in the major areas of applicable mathematics 320........Differential Equations Math 301.......Analysis Math 208 or 316.......Complex Analysis 5 more math courses Two CSC courses or one CSC course and Engr 1 B.S. in Statistics Statistical analysis forms a fundamental tool in all experimental sciences and is important in understanding chance phenomena. Mathematical principles, especially probability theory, underlie all statistical analyses. This program requires 30 hours of Professional Electives to be selected from at least two fields of application of statistics, such as biology, psychology, social relations, computer science, engineering, economics, and management. Required major courses are Major electives (12 credit hours): At least three courses with specific mathematical and statistical content chosen with the approval of the faculty advisor. Minor programs The department offers the following minor programs. Pure Mathematics Applied Mathematics Probability and Statistics Actuarial Science For each program, the requirement is Math 21, 22, and 23, or 31, 32, and 33, plus four courses from a list of specified courses. See the catalogue for the lists of specified courses. In recent years, we have had approximately 10-15 math majors graduating each year. This means that classes are small, so that you get to know your professors and fellow students well. Some of the things that our recent graduates have done after graduation include

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